Evans, S., Hening, A. & Schreiber, S. (submitted), "Protected polymorphisms and evolutionary stability of patchselection strategies in stochastic environments"

BibTeX:
@article{, author = {S.N. Evans and A. Hening and S.J. Schreiber}, title = {Protected polymorphisms and evolutionary stability of patchselection strategies in stochastic environments}, year = {submitted}, url = {http://biorxiv.org/content/early/2014/04/03/003780} } 
Gaylord, B., Kroeker, K., Sunday, J., Anderson, K., Barry, J., Brown, N., Connell, S., Dupont, S., Fabricius, K., HallSpencer, J., Klinger, T., Milazzo, M., Munday, P., Russell, B., Sanford, E., Schreiber, S., Thiyagarahan, V., Vaughan, M., Widdicombe, S. & Harley, C. (In review), "Ocean acidification through the lens of ecological theory", Ecology (invited review).

BibTeX:
@article{, author = {B. Gaylord and K.J. Kroeker and J.M. Sunday and K.M. Anderson and J.P. Barry and N. Brown and S.D. Connell and S. Dupont and K.E. Fabricius and J.M. HallSpencer and T. Klinger and M. Milazzo and P.L. Munday and B.D. Russell and E. Sanford and S.J. Schreiber and V. Thiyagarahan and M. Vaughan and S. Widdicombe and C.D.G. Harley}, title = {Ocean acidification through the lens of ecological theory}, journal = {Ecology (invited review)}, year = {In review} } 
Reigeda, C., Schreiber, S., Altermatt, F. & Holyoak, M. (In revision), "Metapopulation dynamics on ephemeral patches", American Naturalist.

BibTeX:
@article{, author = {Reigeda, C. and Schreiber, S.J. and Altermatt, F and Holyoak, M}, title = {Metapopulation dynamics on ephemeral patches}, journal = {American Naturalist}, year = {In revision} } 
Rosenheim, J., Williams, N. & Schreiber, S.J. (In press), "Parental optimism versus parental pessimism in plants: how common should we expect pollen limitation to be?", American Naturalist.

BibTeX:
@article{amnat14a, author = {J.A. Rosenheim and N.M. Williams and S. J. Schreiber}, title = {Parental optimism versus parental pessimism in plants: how common should we expect pollen limitation to be?}, journal = {American Naturalist}, year = {In press}, url = {http://www.amnat.org/an/newpapers/JulyRosenheim.html} } 
Roth, G. & Schreiber, S.J. (In review), "Pushed to brink: Allee effects, environmental stochasticity, and extinction"

BibTeX:
@article{, author = {Roth, G. and Schreiber, S. J.}, title = {Pushed to brink: Allee effects, environmental stochasticity, and extinction}, year = {In review}, url = {http://biorxiv.org/content/early/2014/05/07/004887} } 
Schreiber, S.J., Rosenheim, J., Williams, N. & Harder, L. (In press), "Evolutionary and Ecological Consequences of Multiscale Variation in Pollen Receipt for Seed Production", American Naturalist.

BibTeX:
@article{amnat14b, author = {Schreiber, S. J. and Rosenheim, J.A. and Williams, N.W. and Harder, L.D.}, title = {Evolutionary and Ecological Consequences of Multiscale Variation in Pollen Receipt for Seed Production}, journal = {American Naturalist}, year = {In press} } 
Roth G and Schreiber S (In press), "Persistence in fluctuating environments for interacting structured populations", Journal of Mathematical Biology.

BibTeX:
@article{, author = {Roth, G. and Schreiber, S.J.}, title = {Persistence in fluctuating environments for interacting structured populations}, journal = {Journal of Mathematical Biology}, year = {In press}, url = {http://arxiv.org/abs/1304.1862} } 
Schreiber, S., Smith, K. & Getz, W. (2014), "Calculus for the Life Sciences", pp. 710. John Wiley & Sons.

BibTeX:
@book{, author = {S.J. Schreiber and K. Smith and W.M. Getz}, title = {Calculus for the Life Sciences}, publisher = {John Wiley and Sons}, year = {2014}, url = {http://www.wiley.com/WileyCDA/WileyTitle/productCdEHEP002970.html} } 
Faure, M. & Schreiber, S. (2014), "Quasistationary distributions for randomly perturbed dynamical systems", Annals of Applied Probability. Vol. 24, pp. 553598.

BibTeX:
@article{, author = {M. Faure and S.J. Schreiber}, title = {Quasistationary distributions for randomly perturbed dynamical systems}, journal = {Annals of Applied Probability}, year = {2014}, volume = {24}, pages = {553598}, url = {http://projecteuclid.org/euclid.aoap/1394465365} } 
Evans SN, Ralph P, Schreiber SJ and Sen A (2013), "Stochastic population growth in spatially heterogeneous environments", Journal of Mathematical Biology. Vol. 66, pp. 423476.

BibTeX:
@article{evansetal12, author = {S. N. Evans and P. Ralph and S. J. Schreiber and A. Sen}, title = {Stochastic population growth in spatially heterogeneous environments}, journal = {Journal of Mathematical Biology}, year = {2013}, volume = {66}, pages = {423476}, url = {http://link.springer.com/article/10.1007%2Fs0028501205140} } 
Schreiber SJ (2013), "Motivating calculus with biology in Undergraduate Mathematics for the Life Sciences: Processes, Models, Assessment, and Directions" , pp. 177188. Mathematical Association of America.

BibTeX:
@inbook{MAA11, author = {S. J. Schreiber}, editor = {J. Carpenter, G. Ledder, and T. Comar}, title = {Motivating calculus with biology in Undergraduate Mathematics for the Life Sciences: Processes, Models, Assessment, and Directions}, publisher = {Mathematical Association of America}, year = {2013}, pages = {177188} } 
Park M, Loverdo C, Schreiber S and LloydSmith J (2013), "Multiple scales of selection influence the evolutionary emergence of novel pathogens", Philiosophical Transactions of the Royal Soceity. B. Vol. 368

BibTeX:
@article{parketal13, author = {M. Park and C. Loverdo and S.J. Schreiber and J. LloydSmith}, title = {Multiple scales of selection influence the evolutionary emergence of novel pathogens}, journal = {Philiosophical Transactions of the Royal Soceity. B}, year = {2013}, volume = {368}, url = {http://rstb.royalsocietypublishing.org/content/368/1614/20120333.abstract} } 
Schreiber SJ and Killingback TP (2013), "Spatial heterogeneity promotes coexistence of rockpaperscissor metacommunities", Theoretical Population Biology. Vol. 86, pp. 111.

BibTeX:
@article{schreiberkillingback13, author = {S. J. Schreiber and T. P. Killingback}, title = {Spatial heterogeneity promotes coexistence of rockpaperscissor metacommunities}, journal = {Theoretical Population Biology}, year = {2013}, volume = {86}, pages = {111}, url = {http://arxiv.org/abs/1207.0485} } 
Schreiber S (2012), "Evolution of patch selection in stochastic environments", American Naturalist. Vol. 180, pp. 1734.

BibTeX:
@article{amnat12, author = {Schreiber, S.J.}, title = {Evolution of patch selection in stochastic environments}, journal = {American Naturalist}, year = {2012}, volume = {180}, pages = {1734}, url = {http://www.jstor.org/discover/10.1086/665655?uid=3739560&uid=2129&uid=2&uid=70&uid=4&uid=3739256&sid=211017262446432} } 
Schreiber SJ (2012), "Persistence for stochastic difference equations: A minireview", Journal of Difference Equations and Applications (Special Issue on Stochastic Difference Equations). Vol. 18, pp. 13811403.

Abstract: Understanding under what conditions populations, whether they be plants,animals, or viral particles, persist is an issue of theoretical and practical importance in population biology. Both biotic interactions and environmental fluctuations are key factors that can facilitate or disrupt persistence. One approach to examining the interplay between these deterministic and stochastic forces is the construction and analysis of stochastic difference equations $X_t+1=F(X_t,t+1)$ where $X_t in represents the state of the populations and $1,2, is a sequence of random variables representing environmental stochasticity. In the analysis of these stochastic models, many theoretical population biologists are interested in whether the models are stochastically bounded and persistent. Stochastic boundedness asserts that asymptotically $X_t$ tends to remain in compact sets. In contrast, stochastic persistence requires that $X_t$ tends to be ``repelled'' by some ''extinction set'' $S_0subset . Here, results on both of these proprieties are reviewed for single species, multiple species, and structured population models. The results are illustrated with applications to stochastic versions of the Hassell and Ricker single species models, Ricker, BevertonHolt, and lottery models of competition, and lottery models of rockpaperscissor games. A variety of conjectures and suggestions for future research are presented. 
BibTeX:
@article{jdea12, author = {S. J. Schreiber}, title = {Persistence for stochastic difference equations: A minireview}, journal = {Journal of Difference Equations and Applications (Special Issue on Stochastic Difference Equations)}, year = {2012}, volume = {18}, pages = {13811403}, url = {http://arxiv.org/abs/1109.5967}, doi = {10.1080/10236198.2011.628662} } 
Loverdo C, Park M, Schreiber SJ and LloydSmtih J (2012), "Influence of viral replication mechanisms on withinhost evolutionary dynamics", Evolution. Vol. 66, pp. 34623471.

BibTeX:
@article{loverdoetal12, author = {C. Loverdo and M. Park and S. J. Schreiber and J.O. LloydSmtih}, title = {Influence of viral replication mechanisms on withinhost evolutionary dynamics}, journal = {Evolution}, year = {2012}, volume = {66}, pages = {34623471}, url = {http://onlinelibrary.wiley.com/doi/10.1111/j.15585646.2012.01687.x/abstract}, doi = {10.1111/j.15585646.2012.01687.x} } 
Schaiber JG, Silverstein R, Kaczmarczyk AN, Rutaganira RU, Aggarwal T, Schwemmer M, Hom CL, Grossberg RK and Schreiber SJ (2012), "Constraints on the use of lifespan shortening Wolbachia to control dengue fever", Journal of Theoretical Biology. Vol. 297, pp. 2632.

Abstract: Dengue fever, a viral disease spread by the mosquito Aedes aegypti affects 50 to 100 million people a year in many tropical countries. Because the virus must incubate within mosquito hosts for two weeks before becoming virulent, shortening the lifespan of mosquitoes may curtail dengue transmission. We developed a continuous time reactiondiffusion model of the spatial spread of Wolbachia through a population of Ae. aegypti. This model incorporates the lifespanshortening effects of Wolbachia on infected Ae. aegypti and the fitness advantage to infected females due to cytoplasmic incompatibility (CI).We found that local establishment of the Wolbachia infection can occur if the fitness advantage due to CI exceeds the fitness reduction due to lifespan shortening effects. However, spatial spread is possible only if the fitness advantage due to CI is twice as great as the fitness reduction due to lifespan shortening effects. Using data from the literature, we estimated all demographic parameters for infected and uninfected mosquitoes and computed the velocities of spread of infection. Our most optimistic estimates suggest that the spatial spread of lifespanshortening Wolbachia may be too slow to effectively control dengue fever. However, as these estimates of demographic parameters may not accurately reflect natural conditions,further research is necessary to corroborate these predictions. 
BibTeX:
@article{schraiberetal12, author = {J. G. Schaiber and R. Silverstein and A. N. Kaczmarczyk and R. U. Rutaganira and T. Aggarwal and M. Schwemmer and C. L. Hom and R. K. Grossberg and S. J. Schreiber}, title = {Constraints on the use of lifespan shortening Wolbachia to control dengue fever}, journal = {Journal of Theoretical Biology}, year = {2012}, volume = {297}, pages = {2632}, url = {http://www.sciencedirect.com/science/article/pii/S0022519311006151}, doi = {10.1016/j.jtbi.2011.12.006} } 
Ellner SP and Schreiber SJ (2012), "Temporally variable dispersal and demography can accelerate the spread of invading species", Theoretical Population Biology. Vol. 82, pp. 283298.

BibTeX:
@article{tpb12, author = {S. P. Ellner and S. J. Schreiber}, title = {Temporally variable dispersal and demography can accelerate the spread of invading species}, journal = {Theoretical Population Biology}, year = {2012}, volume = {82}, pages = {283298}, url = {http://www.sciencedirect.com/science/article/pii/S0040580912000445} } 
Altermatt F, Schreiber S and Holyoak M (2011), "Interactive effects of disturbance and directionality of dispersal on species richness and composition in metacommunities", Ecology. Vol. 92, pp. 859870.

Abstract: Dispersal among ecological communities is usually assumed to be random in direction, or to vary in distance or frequency among species. However, a variety of natural systems and types of organisms may experience dispersal that is biased by directional currents or by gravity on hillslopes. We developed a general model for competing species in metacommunities to evaluate the role of directionallybiased dispersal on species diversity, abundance and traits. We then tested this model using a microcosm experiment with protists and rotifers. Both the model and experiment demonstrated that diversity in local communities was reduced by directionallybiased dispersal, especially that biased away from disturbed patches. In simulations not all initially introduced species could coexist, and biased dispersal reduced regional diversity, but this was not true in the experiment, where species were able to coexist locally in the absence of dispersal. Abundance of species (and composition) in local communities was a product of disturbance intensity but not dispersal, whereas local abundance was influenced by dispersal in some simulations. High disturbance selected for species with high intrinsic growth rates and lower competitive ranking. Overall, our experiment confirmed our model predictions about the key role of dispersal directionality in (meta)community response to disturbance.!! 
BibTeX:
@article{altermattetal11, author = {F. Altermatt and S.J. Schreiber and M. Holyoak}, title = {Interactive effects of disturbance and directionality of dispersal on species richness and composition in metacommunities}, journal = {Ecology}, year = {2011}, volume = {92}, pages = {859870}, url = {http://www.esajournals.org/doi/abs/10.1890/101095.1} } 
Bolnick D, Amarasekare P, M. S. Araújo, R. Bürger J. Levine, Novak M, Rudolf VH, Schreiber S, Urban M and Vasseur D (2011), "Why intraspecific trait variation matters in community ecology", Trends in Ecology and Evolution. Vol. 26, pp. 185194.

Abstract: Natural populations consist of phenotypically diverse individuals that exhibit variation in their demographic parameters and intra and interspeciÞc interactions.Recent experimental work indicates that such variation can have signiÞcant ecological effects. However, ecological models typically disregard this variation and focus instead on trait means and total population density. Under what situations is this simpliÞcation appropriate? Why might intraspeciÞc variation alter ecological dynamics? In this review we synthesize recent theory and identify six general mechanisms by which trait variation changes the outcome of ecological interactions. These mechanisms include several direct effects of trait variation per se and indirect effects arising from the role of genetic variation in trait evolution.!! 
BibTeX:
@article{bolnicketal11, author = {D. Bolnick and P. Amarasekare and M. S. Araújo, and R. Bürger, 
Schreiber SJ, Bolnick D and Bürger R (2011), "The community effects of phenotypic and genetic variation within a predator population", Ecology. Vol. 92, pp. 15821593.

Abstract: Natural populations are heterogeneous mixtures of individuals differing in physiology, morphology, and behavior. Despite the ubiquity of phenotypic variation within natural populations, its effects on the dynamics of ecological communities are not well understood. Here, we use a quantitative genetics framework to examine how phenotypic variation in a predator affects the outcome of apparent competition between its two prey species. Classical apparent competition theory predicts that prey have reciprocally negative effects on each other. The addition of phenotypic trait variation in predation can marginalize these negative effects, mediate coexistence, or generate positive indirect effects between the prey species. Longterm coexistence or facilitation, however, can be preceded by long transients of extinction risk whenever the heritability of phenotypic variation is low. Greater heritability can circumvent these ecological transients, but also can generate oscillatory and chaotic dynamics. These dramatic changes in ecological outcomes, in the sign of indirect effects, and in stability suggest that studies which ignore intraspecific trait variation may reach fundamentally incorrect conclusions regarding ecological dynamics!! 
BibTeX:
@article{ecology11b, author = {S. J. Schreiber and D. Bolnick and R. Bürger}, title = {The community effects of phenotypic and genetic variation within a predator population}, journal = {Ecology}, year = {2011}, volume = {92}, pages = {15821593}, url = {http://www.esajournals.org/doi/abs/10.1890/102071.1} } 
Schreiber SJ and Li CK (2011), "Evolution of unconditional dispersal in periodic environments", Journal of Biological Dynamics (special issue on Adaptive Dynamics). Vol. 5, pp. 120134.

Abstract: Organisms modulate their fitness in heterogeneous environments by dispersing. Prior work shows that there is selection against ``unconditional'' dispersal in spatially heterogeneous environments. ``Unconditional'' means individuals disperse at a rate independent of their location. We prove that if withinpatch fitness varies spatially and between two values temporally, then there is selection for unconditional dispersal: any evolutionarily stable strategy (ESS) or evolutionarily stable coalition (ESC) includes a dispersive phenotype. Moreover, at this ESS or ESC, there is at least one sink patch (i.e. geometric mean of fitness less than one) and no sources patches (i.e. geometric mean of fitness greater than one). These results coupled with simulations suggest that spatialtemporal heterogeneity due to abiotic forcing result in either an ESS with a dispersive phenotype or an ESC with sedentary and dispersive phenotypes. In contrast, spatialtemporal heterogeneity due to biotic interactions select for higher dispersal rates that ultimately spatially synchronize population dynamics. 
BibTeX:
@article{jbd10, author = {S. J. Schreiber and C. K. Li}, title = {Evolution of unconditional dispersal in periodic environments}, journal = {Journal of Biological Dynamics (special issue on Adaptive Dynamics)}, year = {2011}, volume = {5}, pages = {120134}, url = {http://arxiv.org/abs/1007.5267} } 
Schreiber SJ, Benaïm M and Atchadé KAS (2011), "Persistence in fluctuating environments", Journal of Mathematical Biology. Vol. 62, pp. 655683.

Abstract: Understanding under what conditions interacting populations, whether they be plants, animals, or viral particles, coexist is a question of theoretical and practical importance in population biology. Both biotic interactions and environmental fluctuations are key factors that can facilitate or disrupt coexistence. To better understand this interplay between these deterministic and stochastic forces, we develop a mathematical theory extending the nonlinear theory of permanence for deterministic systems to stochastic difference and differential equations. Our condition for coexistence requires that there is a fixed set of weights associated with the interacting populations and this weighted combination of populations' invasion rates is positive for any (ergodic) stationary distribution associated with a subcollection of populations. Here, an invasion rate corresponds to an average percapita growth rate along a stationary distribution. When this condition holds and there is sufficient noise in the system (i.e. all population states are accessible), we show that the populations approach a unique positive stationary distribution. Moreover, we show that our coexistence criterion is robust to small perturbations of the model functions. Using this theory, we illustrate that (i) environmental noise enhances or inhibits coexistence in communities with rockpaperscissor dynamics depending on correlations between interspecific demographic rates, (ii) stochastic variation in mortality rates has no effect on the coexistence criteria for discretetime LotkaVolterra communities, and (iii) random forcing can promote genetic diversity in the presence of exploitative interactions.!! 
BibTeX:
@article{jmb10, author = {S. J. Schreiber and M. Benaïm and K. A. S. Atchadé}, title = {Persistence in fluctuating environments}, journal = {Journal of Mathematical Biology}, year = {2011}, volume = {62}, pages = {655683}, doi = {10.1007/s0028501003495} } 
Schreiber S (2011), "Mathematical Dances with Wolves", Science. Vol. 334, pp. 12141215.

BibTeX:
@article{science11, author = {S.J. Schreiber}, title = {Mathematical Dances with Wolves}, journal = {Science}, year = {2011}, volume = {334}, pages = {12141215}, url = {http://www.sciencemag.org/content/334/6060/1214.summary}, doi = {10.1126/science.1214845} } 
Schreiber SJ and Ryan ME (2011), "Invasion speeds of structured populations in fluctuating environments", Theoretical Ecology. Vol. 4, pp. 423434.

Abstract: We live in a time where climate models predict future increases in environmental variability and biological invasions are becoming increasingly frequent. A key to developing effective responses to biological invasions in increasingly variable environments will be estimates of their rates of spatial spread and the associated uncertainty of these estimates. Using stochastic, stagestructured, integrodifference equation models, we show analytically that invasion speeds are asymptotically normally distributed with a variance that decreases in time. We apply our methods to a simple juvenileadult model with stochastic variation in reproduction and an illustrative example with published data for the perennial herb, Calathea ovandensis. These examples buttressed by additional analysis reveal that increased variability in vital rates simultaneously slow down invasions yet generate greater uncertainty about rates of spatial spread. Moreover, while temporal autocorrelations in vital rates inflate variability in invasion speeds, the effect of these autocorrelations on the average invasion speed can be positive or negative depending on life history traits and how well vital rates ``remember'' the past.!! 
BibTeX:
@article{te11, author = {S. J. Schreiber and M. E. Ryan}, title = {Invasion speeds of structured populations in fluctuating environments}, journal = {Theoretical Ecology}, year = {2011}, volume = {4}, pages = {423434}, url = {http://www.springerlink.com/content/923g010x3447l854/fulltext.pdf}, doi = {10.1007/s1208001000985} } 
Schreiber SJ (2011), "Ordinary Differential equations", In Sourcebook in Theoretical Ecology. Berkeley, California University of California Press.

BibTeX:
@inproceedings{, author = {S. J. Schreiber}, editor = {A. Hastings and L. Gross}, title = {Ordinary Differential equations}, booktitle = {Sourcebook in Theoretical Ecology}, publisher = {University of California Press}, year = {2011} } 
Hofbauer J and Schreiber SJ (2010), "Robust permanence for interacting structured populations", Journal of Differential Equations. Vol. 248, pp. 19551971.

Abstract: The dynamics of interacting structured populations can be modeled by $dx_idt= A_i (x)x_i$ where $x_iin n_i$, $x=(x_1,x_k)$, and $A_i(x)$ are matrices with nonnegative offdiagonal entries. These models are permanent if there exists a positive global attractor and are robustly permanent if they remain permanent following perturbations of $A_i(x)$. Necessary and sufficient conditions for robust permanence are derived using dominant Lyapunov exponents $i($ of the $A_i(x)$ with respect to invariant measures $. The necessary condition requires $i i(>0$ for all ergodic measures with support in the boundary of the nonnegative cone. The sufficient condition requires that the boundary admits a Morse decomposition such that $i i(>0$ for all invariant measures $ supported by a component of the Morse decomposition. When the Morse components are Axiom A, uniquely ergodic, or support all but one population, the necessary and sufficient conditions are equivalent. Applications to spatial ecology, epidemiology, and gene networks are given.!! 
BibTeX:
@article{jde10, author = {J. Hofbauer and S. J. Schreiber}, title = {Robust permanence for interacting structured populations}, journal = {Journal of Differential Equations}, year = {2010}, volume = {248}, pages = {19551971} } 
Edwards KF and Schreiber SJ (2010), "Preemption of space can lead to intransitive coexistence of competitors", Oikos. Vol. 119, pp. 12011209.

Abstract: Intransitive competition has the potential to be a powerful contributor to species coexistence, but there are few proposed biological mechanisms that could create intransitivities in natural communities. Using a threespecies model of competition for space, we demonstrate a mechanism for coexistence that combines a colonizationcompetition tradeoff between two species with the ability of a third species to preempt space from the other competitors. The combination of differential abilities to colonize, preempt, and overtake space creates a community where no single species can exclude both of its competitors. The dynamics of this kind of community are analogous to rockpaperscissors competition, and the threespecies community can persist even though not all pairs of species can coexist in isolation. In distinction to prior results, this is a mechanism of intransitivity that does not require nonhierarchical local interference competition. We present parameter estimates from a subtidal marine community illustrating how documented competitive traits can lead to preemptionbased intransitivities in natural communities, and we describe methods for an empirical test of the occurrence of this mechanism. 
BibTeX:
@article{oikos10, author = {K. F. Edwards and S. J. Schreiber}, title = {Preemption of space can lead to intransitive coexistence of competitors}, journal = {Oikos}, year = {2010}, volume = {119}, pages = {12011209} } 
Schreiber SJ (2010), "Interactive effects of temporal correlations, spatial heterogeneity, and dispersal on population persistence", Proceedings of the Royal Society: Biological Sciences. Vol. 277, pp. 19071914.

Abstract: It is an ecological truism that population persistence depends on a population's growth rate when rare. To understand the interplay between temporal correlations, spatial heterogeneity, and dispersal on persistence, an analytic approximation for this growth rate is derived for partially mixing populations. Partial mixing has two effects on population growth. In the absence of temporal correlations in relative fitness, greater movement to patches with, on average, higher relative fitness increases population growth rates. In the absence of spatial heterogeneity in the average relative fitnesses, lower dispersal rates enhance growth rates when temporal correlations of relative fitness within a patch exceed temporal correlations in relative fitness between different patches. This approximation implies that metapopulations whose expected fitness in every patch is less than one can persist if there are positive temporal correlations in relative fitness, sufficiently weak spatial correlations, and sufficiently low dispersal rates. It also implies that movement into lowerquality habitats is optimal whenever the net variation in percapita growth rates is sufficiently larger than the mean difference in percapita growth rates. Moreover, temporal correlations, whether they be negative or positive, can enhance population growth for optimal dispersal rates.!! 
BibTeX:
@article{prsb10, author = {S. J. Schreiber}, title = {Interactive effects of temporal correlations, spatial heterogeneity, and dispersal on population persistence}, journal = {Proceedings of the Royal Society: Biological Sciences}, year = {2010}, volume = {277}, pages = {19071914} } 
Schreiber SJ and *Saltzman E (2009), "Evolution of predator and prey movement into sink habitats", American Naturalist. Vol. 174, pp. 6881.

Abstract: Mathematical models of predatorprey interactions in a patchy landscape are used to explore the evolution of dispersal into sink habitats. When evolution is constrained to a single trophic level, three evolutionary outcomes are observed. If predatorprey dynamics are sufficiently stable in source habitats, then there is an evolutionarily stable strategy (ESS) corresponding to sedentary phenotypes that specialize on source habitats. However, if predatorprey dynamics are sufficiently unstable in source habitats, then either an ESS corresponding to dispersive phenotypes utilizing both source and sink habitats or an evolutionary stable coalition (ESC) between dispersive and sedentary phenotypes emerges. ESCs only occur if dispersal into sink habitats can stabilize the predatorprey interactions. When evolution of dispersal proceeds at both trophic levels, nine evolutionary outcomes corresponding to any combination of specialists and generalists at one or both trophic levels were observed. Coevolution is largely topdown driven. If the predator mortality rate in sink habitats is low, then selection pressure for predator movement into sink habitats can forestall the evolution of prey sink populations. Alternatively, if this mortality rate is high, then the predators ultimately play a sedentary ESS. Only at intermediate predator mortality rates is there selection for predator and prey movement into sink habitats. These results suggest that the instability of predatorprey interactions may foster speciation near species borders. Moreover, they illustrate an evolutionary paradox of enrichment in which enriching source habitats can result in a loss of phenotypic diversity. !! 
BibTeX:
@article{amnat09a, author = {S. J. Schreiber and E. *Saltzman}, title = {Evolution of predator and prey movement into sink habitats}, journal = {American Naturalist}, year = {2009}, volume = {174}, pages = {6881} } 
Schreiber SJ and LloydSmith JO (2009), "Invasion dynamics in spatially heterogenous environments", American Naturalist. Vol. 174, pp. 490505.

Abstract: Biological invasions, including infectious disease outbreaks and biocontrol introductions, often involve small numbers of individuals arriving in spatially heterogeneous environments. Small numbers lead to demographic stochasticity, and spatial heterogeneity means that establishment success depends critically on the introduction sites and movement patterns of invaders. We present a general stochastic modeling framework to address how spatial heterogeneity and movement patterns determine establishment success, population growth, and rates of spatial spread. For dispersallimited populations, our analysis reveals that spatial heterogeneity increases the expected population growth rate and that local reproductive numbers determine establishment success. Higher dispersal rates decrease the expected population growth rate, but can enhance establishment success, particularly when movement patterns are positively correlated with local reproductive numbers. We also find that several small, randomly distributed propagules of invaders are more likely to succeed than a single large propagule. Even if invasions are ultimately successful, there may be substantial time lags before an invader reaches observable densities. These time lags are longer for invasions into patches where extinction risk is high and in landscapes where metapopulationscale population growth rate is low, while the opposite holds true for rates of spatial spread. Sensitivity analysis of our models provides guidance for control efforts.!! 
BibTeX:
@article{amnat09b, author = {S. J. Schreiber and J. O. LloydSmith}, title = {Invasion dynamics in spatially heterogenous environments}, journal = {American Naturalist}, year = {2009}, volume = {174}, pages = {490505} } 
Kon R and Schreiber SJ (2009), "Host and multiple parasitoid dynamics with egg limitation", SIAM Journal of Applied Mathematics. Vol. 69, pp. 959976.

Abstract: To address the contentious issue of multiple parasitoid introductions in classical biological control, a discretetime model of multiparasitoidhost interactions that accounts for host densitydependence and egg limitation is introduced and analyzed. For parasitoids that are egglimited but not searchlimited, the model is proven to exhibit four types of dynamics: host failure in which the host goes extinct in the presence or absence of the parasitoids, parasitoid driven extinction in which the parasitoid complex invariably drives the host extinct, host persistence, and conditional host persistence in which depending on the initial ratios of host to parasitoid densities the host is either driven extinct or persists. In the case of host persistence, the dynamics of the system are shown to be asymptotic to the dynamics of an appropriately defined onedimensional difference equation. The results illustrate how the establishment of one or more parasitoids can facilitate the invasion of another parasitoid and how a complex of parasitoids can drive a host extinct despite every species in the complex being unable to do so. The effects of including search limitation are also explored. 
BibTeX:
@article{siap09, author = {R. Kon and S. J. Schreiber}, title = {Host and multiple parasitoid dynamics with egg limitation}, journal = {SIAM Journal of Applied Mathematics}, year = {2009}, volume = {69}, pages = {959976} } 
Benaïm M and Schreiber SJ (2009), "Persistence of structured populations in random environments", Theoretical Population Biology. Vol. 76, pp. 1934.

Abstract: Environmental fluctuations often have different impacts on individuals that differ in size, age, or spatial location. To understand how population structure, environmental fluctuations, and densitydependent interactions influence population dynamics, we provide a general theory for persistence for densitydependent matrix models in random environments. For populations with compensating density dependence, exhibiting ``bounded'' dynamics, and living in a stationary environment, we show that persistence is determined by the stochastic growth rate (alternatively, dominant Lyapunov exponent) when the population is rare. If this stochastic growth rate is negative, then the total population abundance goes to zero with probability one. If this stochastic growth rate is positive, there is a unique positive stationary distribution. Provided there are initially some individuals in the population, the population converges in distribution to this stationary distribution and the empirical measures almost surely converge to the distribution of the stationary distribution. For models with overcompensating densitydependence, weaker results are proven. Methods to estimate stochastic growth rates are presented. To illustrate the utility of these results, applications to unstructured, spatially structured, and stagestructured population models are given. For instance, we show that diffusively coupled sink populations can persist provided that within patch fitness is sufficiently variable in time but not strongly correlated across space. !! 
BibTeX:
@article{tpb09, author = {M. Benaïm and S. J. Schreiber}, title = {Persistence of structured populations in random environments}, journal = {Theoretical Population Biology}, year = {2009}, volume = {76}, pages = {1934} } 
Schreiber SJ and Rudolf V (2008), "Crossing habitat boundaries: Coupling dynamics of ecosystems through complex life cycles", Ecology Letters. Vol. 11, pp. 576587.

Abstract: Ecosystems are often indirectly connected through consumers with complex life cycles (CLC), in which different life stages inhabit different ecosystems. Using a structured consumer resource model that accounts for the independent effects of two resources on consumer growth and reproductive rates, we show that such indirect connections between ecosystems can result in alternative stable states characterized by adult dominated and juvenile dominated consumer populations. As a consequence, gradual changes in ecosystem productivity or mortality rates of the consumer can lead to dramatic and abrupt regime shifts across different ecosystems, hysteresis, and counterintuitive changes in the consumer abundances. Whether these counterintuitive or abrupt responses occur depend on the relative productivity of both habitats and which consumer lifestage inhabits the manipulated ecosystem. These results demonstrate the strong yet complex interactions between ecosystems coupled through consumers with CLC and the need to think across ecosystems to reliably predict the consequences of natural or anthropogenic changes. !! 
BibTeX:
@article{ecolets08, author = {S. J. Schreiber and V. Rudolf}, title = {Crossing habitat boundaries: Coupling dynamics of ecosystems through complex life cycles}, journal = {Ecology Letters}, year = {2008}, volume = {11}, pages = {576587} } 
Lipcius RN, Eggleston DB, Schreiber SJ, Seitz RD, Shen J, Sisson M, Stockhausen WT and Wang HV (2008), "Metapopulation connectivity and stock enhancement of marine species", Reviews in Fishery Science. Vol. 16, pp. 101110.

Abstract: Various biophysical systems exhibit characteristics of metapopulation and network structure. The specific type of metapopulation or network structure can have substantially different effects on metapopulation dynamics of marine species with open populations displaying varying degrees of connectivity between subpopulations, and thus can have major consequences on stock enhancement efforts. We investigate the role of connectivity in metapopulation dynamics of the blue crab, Callinectes sapidus, and the Eastern oyster, Crassostrea virginica, with three dimensional hydrodynamic models simulating advection and diffusion. In the case of the blue crab, we model a metapopulation comprised of primary (i.e. seagrass beds) and secondary (i.e. salt marsh fringed coves and shorelines) nursery habitats, and spatially distinct spawning grounds connected via migration corridors. In the model simulations, we distinguish nursery habitats that are recruitment limited, and therefore optimal candidates for stock enhancement through release of hatcheryreared or translocated juveniles. In the case of the Eastern oyster, we model connectivity between numerous oyster reefs (i.e., subpopulations) positioned according to historical observations. Model simulations produce estimates of the degree of connectivity between all pairs of oyster reefs, which subsequently permits assessment of the diversity of patterns in network connectivity. From these results we distinguish the major characteristic types of connectivity patterns among oyster reefs, we identify those reefs most suitable for broodstock enhancement, and we discuss the means by which oyster reef networks, such as those occurring throughout Chesapeake Bay, may be enhanced successfully. 
BibTeX:
@article{rfs08, author = {R.~N. Lipcius and D.~B. Eggleston and S.~J. Schreiber and R.~D. Seitz and J.~Shen and M.~Sisson and W.~T. Stockhausen and H.~V. Wang}, title = {Metapopulation connectivity and stock enhancement of marine species}, journal = {Reviews in Fishery Science}, year = {2008}, volume = {16}, pages = {101110}, url = {http://dx.doi.org/10.1080/10641260701812574} } 
Schreiber SJ (2007), "On persistence and extinction of randomly perturbed dynamical systems", Discrete and Continous Dynamical Systems B. Vol. 7, pp. 457463.

Abstract: Let f:M>M be a continuous map of a locally compact metric space. Models of interacting populations often have a closed invariant set E that corresponds to the loss or extinction of one or more populations. The dynamics of f subject to bounded random perturbations for which E is absorbing are studied. When these random perturbations are sufficiently small, almost sure absorbtion (i.e. extinction) for all initial conditions is shown to occur if and only if M E contains no attractors for f. Applications to evolutionary bimatrix games and uniform persistence are given. In particular, it shown that random perturbations of evolutionary bimatrix game dynamics result in almost sure extinction of one or more strategies.!! 
BibTeX:
@article{dcds07, author = {S. J. Schreiber}, title = {On persistence and extinction of randomly perturbed dynamical systems}, journal = {Discrete and Continous Dynamical Systems B}, year = {2007}, volume = {7}, pages = {457463} } 
Schreiber SJ (2007), "Periodicity, persistence, and collapse in hostparasitoid systems with egg limitation", Journal of Biological Dynamics. Vol. 1, pp. 273  287.

Abstract: There is an emerging consensus that parasitoids are limited by the number of eggs which they can lay as well as the amount of time they can search for their hosts. Since egg limitation tends to destabilize hostparasitoid dynamics, successful control of insect pests by parasitoids requires additional stabilizing mechanisms such as heterogeneity in the distribution of parasitoid attacks and host densitydependence. To better understand how egg limitation, search limitation, heterogeneity in parasitoid attacks, and host densitydependence influence hostparasitoid dynamics, discrete time models accounting for these factors are analyzed. When parasitoids are purely egglimited, a complete anaylsis of the hostparasitoid dynamics are possible. The analysis implies that the parasitoid can invade the host system only if the parasitoid's intrinsic fitness exceeds the host's intrinsic fitness. When the parasitoid can invade, there is a critical threshold, CV*>1, of the coefficient of variation (CV) of the distribution of parasitoid attacks that determines that outcome of the invasion. If parasitoid attacks sufficiently aggregated (i.e., CV>CV*), then the host and parasitoid coexist. Typically (in a topological sense), this coexistence is shown to occur about a periodic attractor or a stable equilibrium. If the parasitoid attacks are sufficiently random (i.e. CV 
BibTeX:
@article{jbd07, author = {Schreiber, S. J.}, title = {Periodicity, persistence, and collapse in hostparasitoid systems with egg limitation}, journal = {Journal of Biological Dynamics}, year = {2007}, volume = {1}, pages = {273  287} } 
LloydSmith JO, Schreiber SJ and Getz WM (2006), "Moving beyond averages: individuallevel variation in disease transmission", In Mathematical studies on human disease dynamics. Providence, RI Vol. 410, pp. 235258. Amer. Math. Soc..

Abstract: It is common practice in disease modeling studies to characterize groups or subgroups using populationaverage parameters, most importantly the basic reproductive number, R0 . This approach overlooks variation at the individual level, which is caused by many factors. In this paper we show evidence of significant individuallevel variation in transmission patterns for several diseases, and discuss how this can be incorporated into epidemiological models. We introduce a natural generalization of R0 : the Ôindividual reproductive numberÕ, v, which is the expected number of secondary cases caused by a given infected individual. Individual reproductive numbers for a population are drawn from a continuous probability distribution with mean equal to R0 (or to the effective reproductive number, R, if the population is not wholly susceptible). In this framework, superspreading events correspond to extreme values from the righthand tail of the distribution of v, and we propose a precise and generalizable deÞnition of superspreading events based on probabilistic considerations. We analyze detailed transmission data for a range of directlytransmitted diseases, and Þnd that conventional models assuming homogeneous transmission cannot account for observed patterns. Analysis of a branching process model incorporating individuallevel heterogeneity reveals that observed levels of variation cause invasion dynamics to differ dramatically from predictions based on population averages. We explore the implications of these Þndings for outbreak control policies, demonstrating that individualspeciÞc control measures are more likely to stop an outbreak than populationwide measures when both have the same effect on R0 . We also highlight the effectiveness of measures targeting highly infectious individuals, and discuss how our results relate to recentlyproposed surveillance methods for emerging diseases. We conclude by discussing future challenges in empirical and theoretical studies. 
BibTeX:
@incollection{book06, author = {J. O. LloydSmith and S. J. Schreiber and W. M. Getz}, title = {Moving beyond averages: individuallevel variation in disease transmission}, booktitle = {Mathematical studies on human disease dynamics}, publisher = {Amer. Math. Soc.}, year = {2006}, volume = {410}, pages = {235258} } 
Schreiber SJ (2006), "Hostparasitoid dynamics of a generalized Thompson model", J. Math. Biol.. Vol. 52, pp. 719732.

Abstract: A discretetime hostparasitoid model including hostdensity dependence and a generalized Thompson escape function is analyzed. This model assumes that parasitoids are egglimited but not searchlimited, and is proven to exhibit five types of dynamics: host failure in which the host goes extinct in the parasitoid's presence or absence, unconditional parasitoid failure in which the parasitoid always goes extinct while the host persists, conditional parasitoid failure in the host and the parasitoid go extinct or coexist depending on the initial hostparasitoid ratio, parasitoid driven extinction in which the parasitoid invariably drives the host to extinction, and coexistence in which the host and parasitoid coexist about a global attractor. The latter two dynamics only occur when the parasitoid's maximal rate of growth exceeds the host's maximal rate of growth. Moreover, coexistence requires parasitism events to be sufficiently aggregated. Small additive noise is proven to alter the dynamical outcomes in two ways. The addition of noise to parasitoid driven extinction results in random outbreaks of the host and parasitoid with varying intensity. Additive noise converts conditional parasitoid failure to unconditional parasitoid failure. Implications for classical biological control are discussed. 
BibTeX:
@article{jmb06, author = {S. J. Schreiber}, title = {Hostparasitoid dynamics of a generalized Thompson model}, journal = {J. Math. Biol.}, year = {2006}, volume = {52}, pages = {719732} } 
Schreiber SJ (2006), "Persistence despite perturbations for interacting populations", Journal of Theoretical Biology. Vol. 242, pp. 84452.

Abstract: Two definitions of persistence despite perturbations in deterministic models are presented. The first definition, persistence despite frequent small perturbations, is shown to be equivalent to the existence of a positive attractor i.e. an attractor bounded away from extinction. The second definition, persistence despite rare large perturbations, is shown to be equivalent to permanence i.e. a positive attractor whose basin of attraction includes all positive states. Both definitions set up a natural dichotomy for classifying models of interacting populations. Namely, a model is either persistent despite perturbations or not. When it is not persistent, it follows that all initial conditions are prone to extinction due to perturbations of the appropriate type. For frequent small perturbations, this method of classification is shown to be generically robust: there is a dense set of models for which persistent (respectively, extinction prone) models lies within an open set of persistent (resp. extinction prone) models. For rare large perturbations, this method of classification is shown not to be generically robust. Namely, work of Josef Hofbauer and the author have shown there are open sets of ecological models containing a dense sets of permanent models and a dense set of extinction prone models. The merits and drawbacks of these different definitions are discussed. 
BibTeX:
@article{jtb06, author = {S. J. Schreiber}, title = {Persistence despite perturbations for interacting populations}, journal = {Journal of Theoretical Biology}, year = {2006}, volume = {242}, pages = {84452} } 
Li CK and Schreiber SJ (2006), "On dispersal and population growth for multistate matrix models", Linear Algebra Appl.. Vol. 418(23), pp. 900912.

Abstract: To describe the dynamics of stagestructured populations with m stages living in n patches, we consider matrix models of the form SD, where S is a block diagonal matrix with n x n column substochastic matrices S1,..., Sm along the diagonal and D is a block matrix whose blocks are n x n nonnegative diagonal matrices. The matrix S describes movement between patches and the matrix D describes growth and reproduction within the patches. Consider the multiple arc directed graph G consisting of the directed graphs corresponding to the matrices S1,..., Sm, where each directed graph is drawn in a different color. We say G has a polychromatic cycle if G has a directed cycle that includes arcs of more than one color. We prove that p(SD) < p(D) for all block matrices D with nonnegative diagonal blocks if and only if G has no polychromatic cycle. Applications to ecological models are presented. 
BibTeX:
@article{laa06, author = {C. K. Li and S. J. Schreiber}, title = {On dispersal and population growth for multistate matrix models}, journal = {Linear Algebra Appl.}, year = {2006}, volume = {418}, number = {23}, pages = {900912} } 
Schreiber SJ, Lipcius R, Seitz R and Long C (2006), "Dancing between the devil and the deep blue sea: The stabilizing effect of enemyfree sinks and victimless sinks", Oikos. Vol. 113, pp. 6781.

Abstract: Theoretical and empirical studies have shown that enemyvictim interactions in spatially homogenous environments can exhibit diverging oscillations which result in the extinction of one or both species. For enemyvictim models with overlapping generations, we investigate the dynamical implications of spatial heterogeneity created by enemyfree sinks or victimless sinks. An enemyfree sink is a behavioral, physiological or ecological state that reduces or eliminates the victim's vulnerability to the enemy but cannot sustain the victim population. For victims that move in an idealfree manner, we prove that the inclusion of an enemyfree sink shifts the population dynamics from diverging oscillations to stable oscillations. During these stable oscillations, the victim disperses in an oscillatory manner between the enemyfree sink and the enemyoccupied patch. Enemyfree sinks with lower mortality rates exhibit oscillations with smaller amplitudes and longer periods. A victimless sink, on the other hand, is a behavioral, physiological or ecological state in which the enemy has limited (or no) access to its victims. For enemies that move in an idealfree manner, we prove that victimless sinks also stabilize diverging oscillations. Simulations suggest that suboptimal behavior due to information gathering or learning limitations amplify oscillations for systems with enemyfree sinks and dampen oscillations for systems with victimless sinks. These results illustrate that the coupling of a sink created by unstable enemyvictim interactions and a sink created by unsuitable environmental conditions can result in population persistence at the landscape level. 
BibTeX:
@article{oikos06, author = {S. J. Schreiber and R. Lipcius and R. Seitz and C. Long}, title = {Dancing between the devil and the deep blue sea: The stabilizing effect of enemyfree sinks and victimless sinks}, journal = {Oikos}, year = {2006}, volume = {113}, pages = {6781} } 
Schreiber SJ and *Vejdani M (2006), "Handling time promotes the coevolution of aggregation in predatorprey systems", Proceedings of the Royal Society: Biological Sciences. Vol. 273, pp. 185191.

Abstract: Predators often have type II functional responses and live in environments where their life history traits as well as those of their prey vary from patch to patch. To understand how spatial heterogeneity and predator handling times influence the coevolution of patch preferences and ecological stability, we perform an ecological and evolutionary analysis of a NicholsonBailey type model. We prove that coevolutionarily stable prey and searching predators prefer patches that in isolation support higher prey and searching predator densities, respectively. Using this fact, we determine how environmental variation and predator handling times influence the spatial patterns of patch preferences, population abundances and percapita predation rates. In particular, long predator handling times are shown to result in the coevolution of predator and prey aggregation. An analytic expression characterizing ecological stability of the coevolved populations is derived. This expression implies that contrary to traditional theoretical expectations, predator handling time can stabilize predatorprey interactions through its coevolutionary influence on patch preferences. These results are shown to have important implications for classical biological control. !! 
BibTeX:
@article{prsb06, author = {Schreiber, S. J. and M. *Vejdani}, title = {Handling time promotes the coevolution of aggregation in predatorprey systems}, journal = {Proceedings of the Royal Society: Biological Sciences}, year = {2006}, volume = {273}, pages = {185191} } 
Jacobs F and Schreiber SJ (2006), "Random perturbations of dynamical systems with absorbing states", SIAM Journal of Applied Dynamical Systems. Vol. 5, pp. 293312.

BibTeX:
@article{siads06, author = {F. Jacobs and S. J. Schreiber}, title = {Random perturbations of dynamical systems with absorbing states}, journal = {SIAM Journal of Applied Dynamical Systems}, year = {2006}, volume = {5}, pages = {293312} } 
Kirkland S, Li C and Schreiber SJ (2006), "On the evolution of dispersal in patchy landscapes", SIAM Journal on Applied Mathematics. Vol. 66(4), pp. 13661382.

Abstract: To better understand the evolution of dispersal in spatially heterogeneous landscapes, we study difference equation models of populations that reproduce and disperse in a landscape consisting of k patches. The connectivity of the patches and costs of dispersal are determined by a k x k column substochastic matrix S, where Sij represents the fraction of dispersing individuals from patch j that end up in patch i. Given S, a dispersal strategy is a k x 1 vector whose ith entry gives the probability pi that individuals disperse from patch i. If all of the pi's are the same, then the dispersal strategy is called unconditional; otherwise it is called conditional. For two competing populations of unconditional dispersers, we prove that the slower dispersing population (i.e., the population with the smaller dispersal probability) displaces the faster dispersing population. Alternatively, for populations of conditional dispersers without any dispersal costs (i.e., S is column stochastic and all patches can support a population), we prove that there is a one parameter family of strategies that resists invasion attempts by all other strategies.!! 
BibTeX:
@article{siap06, author = {S. Kirkland and C.K. Li and S. J. Schreiber}, title = {On the evolution of dispersal in patchy landscapes}, journal = {SIAM Journal on Applied Mathematics}, year = {2006}, volume = {66}, number = {4}, pages = {13661382} } 
Schreiber SJ and *Kelton M (2005), "Sink habitats can alter ecological outcomes for competing species", Journal of Animal Ecology. Vol. 74(6), pp. 9951004.

Abstract: 1. Species often compete for breeding sites in heterogeneous landscapes consisting of sources and sinks. To understand how the presence or absence of sink breeding sites influence ecological outcomes, we extend Pulliam's sourceÐsink model to competing species. 2. In a homogeneous landscape consisting of source sites, we prove that one species, the 'superior' competitor, competitively excludes the other. Dominance is determined by a simple rule: the species that at equilibrium acquires new breeding sites at a faster rate dominates. 3. We prove that the inclusion of sink sites can alter this ecological outcome by either mediating coexistence, reversing competitive dominance, or facilitating a priority effect. 4. Sinkmediated coexistence requires the species to exhibit asymmetries in acquiring sink sites, the 'inferior' species to have a competitive advantage on sink sites and the ratio of sink to source sites be sufficiently low. 5. For example, if the sink breeding sites are competitive refuges for the 'inferior' competitor and not too low in quality, coexistence occurs if the number of sink sites lies below a threshold. Alternatively, when the number of sink sites exceeds this threshold, competitive dominance is reversed and the 'superior' competitor is displaced. 6. Counterintuitively, despite being unable to support species in isolation, sink habitats embedded in a geographical mosaic of sources and sinks can enhance biodiversity by mediating coexistence or alter species composition by reversing competitive interactions. 
BibTeX:
@article{jae05, author = {S. J. Schreiber and M. *Kelton}, title = {Sink habitats can alter ecological outcomes for competing species}, journal = {Journal of Animal Ecology}, year = {2005}, volume = {74}, number = {6}, pages = {9951004}, url = {http://www.blackwellsynergy.com/doi/abs/10.1111/j.13652656.2005.00996.x}, doi = {10.1111/j.13652656.2005.00996.x} } 
Ruggieri E and Schreiber SJ (2005), "The dynamics of the SchoenerPolisHolt model of intraguild predation", Math. Biosci. Eng.. Vol. 2(2), pp. 279288.

Abstract: Intraguild predation occurs when one species (the intraguild predator) predates on and competes with another species (the intraguild prey). A classic model of this interaction was introduced by Gary Polis and Robert Holt building on a model of competing species by Thomas Schoener. A global analysis reveals that this model exhibits generically six dynamics: extinction of one or both species; coexistence about a globally stable equilibrium; contingent exclusion in which the first established species prevents the establishment of the other species; contingent coexistence in which coexistence or displacement of the intraguild prey depend on initial conditions; exclusion of the intraguild prey; and exclusion of the intraguild predator. Implications for biological control and community ecology are discussed 
BibTeX:
@article{mbe05, author = {E. Ruggieri and S. J. Schreiber}, title = {The dynamics of the SchoenerPolisHolt model of intraguild predation}, journal = {Math. Biosci. Eng.}, year = {2005}, volume = {2}, number = {2}, pages = {279288} } 
LloydSmith J, Schreiber SJ, Kopp PE and Getz WM (2005), "Superspreading and the impact of individual variation on disease emergence", Nature. , pp. 355359.

Abstract: Populationlevel analyses often use average quantities to describe heterogeneous systems, particularly when variation does not arise from identifiable groups. A prominent example, central to our current understanding of epidemic spread, is the basic reproductive number, R0, which is defined as the mean number of infections caused by an infected individual in a susceptible population. Population estimates of R0 can obscure considerable individual variation in infectiousness, as highlighted during the global emergence of severe acute respiratory syndrome (SARS) by numerous `superspreading events' in which certain individuals infected unusually large numbers of secondary cases. For diseases transmitted by nonsexual direct contacts, such as SARS or smallpox, individual variation is difficult to measure empirically, and thus its importance for outbreak dynamics has been unclear. Here we present an integrated theoretical and statistical analysis of the influence of individual variation in infectiousness on disease emergence. Using contact tracing data from eight directly transmitted diseases, we show that the distribution of individual infectiousness around R0 is often highly skewed. Model predictions accounting for this variation differ sharply from averagebased approaches, with disease extinction more likely and outbreaks rarer but more explosive. Using these models, we explore implications for outbreak control, showing that individualspecific control measures outperform populationwide measures. Moreover, the dramatic improvements achieved through targeted control policies emphasize the need to identify predictive correlates of higher infectiousness. Our findings indicate that superspreading is a normal feature of disease spread, and to frame ongoing discussion we propose a rigorous definition for superspreading events and a method to predict their frequency. !! 
BibTeX:
@article{nature05, author = {J. LloydSmith and S. J. Schreiber and P. E. Kopp and W. M. Getz}, title = {Superspreading and the impact of individual variation on disease emergence}, journal = {Nature}, year = {2005}, pages = {355359} } 
Keagy J, Schreiber SJ and Cristol DA (2005), "Replacing Sources with Sinks: When Do Populations Go Down the Drain?", Restoration Ecology. Vol. 13(3), pp. 529535.

Abstract: Permits to destroy wetlands often require the creation of the same type of wetland elsewhere. An assumption underlying this practice is that such created wetlands will replace the ecological functions lost when the developed wetland was destroyed. Part of this ecological function is providing habitat for wildlife, including, in coastal areas, a suite of bird species tied to salt marshes for some portion of their life cycle. We tested the hypothesis that created wetlands provide habitat for the avian communities lost when wetlands are destroyed by comparing the breeding and wintering birds on 11 small created salt marshes with those on 11 natural reference salt marshes that were carefully matched for size and surrounding land cover. We found that, during the breeding season, created salt marshes had lower avian abundance and richness than reference salt marshes. In particular, wetlanddependent species were poorly represented on created wetlands. On the other hand, bird use outside of the breeding season and use by an important salt marsh obligate species, the clapper rail (Rallus longirostris), did not differ. Created wetlands that we surveyed failed to completely replicate the bird and plant communities that we observed on nearby natural reference salt marshes, raising the question of whether current mitigation policies that encourage wetland creation should continue without further investigation into the success of such wetlands at recreating wildlife habitat. 
BibTeX:
@article{restoration05, author = {J. Keagy and S. J. Schreiber and D. A. Cristol}, title = {Replacing Sources with Sinks: When Do Populations Go Down the Drain?}, journal = {Restoration Ecology}, year = {2005}, volume = {13}, number = {3}, pages = {529535} } 
Benaïm M, Schreiber SJ and Tarrés P (2004), "Generalized urn models of evolutionary processes", Annals of Applied Probability. Vol. 14, pp. 14551478.

Abstract: Generalized P—lya urn models can describe the dynamics of finite populations of interacting genotypes. Three basic questions these models can address are: Under what conditions does a population exhibit growth? On the event of growth, at what rate does the population increase? What is the longterm behavior of the distribution of genotypes? To address these questions, we associate a mean limit ordinary differential equation (ODE) with the urn model. Previously, it has been shown that on the event of population growth, the limiting distribution of genotypes is a connected internally chain recurrent set for the mean limit ODE. To determine when growth and convergence occurs with positive probability, we prove two results. First, if the mean limit ODE has an ÒattainableÓ attractor at which growth is expected, then growth and convergence toward this attractor occurs with positive probability. Second, the population distribution almost surely does not converge to sets where growth is not expected and almost surely does not converge to ÒnondegenerateÓ unstable equilibria or periodic orbits of the mean limit ODE. Applications to stochastic analogs of the replicator equations and fertilityselection equations of population genetics are given. !! 
BibTeX:
@article{aap04, author = {M. Benaïm and S. J. Schreiber and P. Tarrés}, title = {Generalized urn models of evolutionary processes}, journal = {Annals of Applied Probability}, year = {2004}, volume = {14}, pages = {14551478} } 
Schreiber SJ (2004), "Coexistence for species sharing a predator", Journal of Differential Equations. Vol. 196(1), pp. 209225.

Abstract: A class of equations describing the dynamics of two prey sharing a common predator are considered. Even though the boundary and internal dynamics can exhibit oscillatory behavior, it is shown these equations are permanent if only if they admit a positive equilibrium. Going beyond permanence, a subclass of equations are constructed that are almost surely permanent but not permanent; there exists an attractor in the positive orthant that attracts Lebesgue almost every (but not every) initial condition. 
BibTeX:
@article{jde04, author = {Schreiber, Sebastian J.}, title = {Coexistence for species sharing a predator}, journal = {Journal of Differential Equations}, year = {2004}, volume = {196}, number = {1}, pages = {209225} } 
Hofbauer J and Schreiber SJ (2004), "To persist or not to persist?", Nonlinearity. Vol. 17, pp. 13931406.

Abstract: Ecological vector fields dot x_i = x_if_i(x) on the nonnegative cone bf R^n_+ on Rn are often used to describe the dynamics of n interacting species. These vector fields are called permanent (or uniformly persistent) if the boundary partial bf R^n_+ of the nonnegative cone is repelling. We construct an open set of ecological vector fields containing a dense subset of permanent vector fields and containing a dense subset of vector fields with attractors on partial bf R^n_+ . In particular, this construction implies that robustly permanent vector fields are not dense in the space of permanent vector fields. Hence, verifying robust permanence is important. We illustrate this result with ecological vector fields involving five species that admit a heteroclinic cycle between two equilibria and the Hastings Powell teacup attractor. !! 
BibTeX:
@article{nonlinearity04, author = {Hofbauer, J. and Schreiber, S. J.}, title = {To persist or not to persist?}, journal = {Nonlinearity}, year = {2004}, volume = {17}, pages = {13931406} } 
Schreiber SJ and *Rittenhouse S (2004), "From simple rules to cycling in community assembly", Oikos. Vol. 105, pp. 349358.

Abstract: Simulation studies of community assembly have frequently observed two related phenomena: (1) the humpty dumpty effect in which communities can not be reconstructed by sequential invasions (i.e. single species invasions separated by long intervals of time) and (2) cycling between subcommunities. To better understand the mechanisms underlying these phenomena, we analyze a system consisting of two predators and two prey competing for a shared resource. We show how simple dominance rules (i.e. R and P rules) lead to cycling between subcommunities consisting of predatorprey pairs; predator and prey invasions alternatively lead to prey displacement via apparent competition and predator displacement via exploitative competition. We also show that these cycles are often dynamically unstable in the population phase space. More specifically, while for too slow invasion rates (i.e. sequential invasions) the system cycles indefinitely, faster invasion rates lead to coexistence of all species. In the later case, the assembly dynamics exhibit transient cycling between predatorprey subcommunities and the length of these transients decreases with the invasion rate and increases with habitat productivity. 
BibTeX:
@article{oikos04, author = {S. J. Schreiber and S. *Rittenhouse}, title = {From simple rules to cycling in community assembly}, journal = {Oikos}, year = {2004}, volume = {105}, pages = {349358} } 
Schreiber SJ (2004), "On Allee effects in structured populations", Proc. Amer. Math. Soc.. Vol. 132(10), pp. 30473053 (electronic).

BibTeX:
@article{pams04, author = {S. J. Schreiber}, title = {On Allee effects in structured populations}, journal = {Proc. Amer. Math. Soc.}, year = {2004}, volume = {132}, number = {10}, pages = {30473053 (electronic)} } 
Schreiber SJ and *Tobiason GA (2003), "The evolution of resource use", J. Math. Biol.. Vol. 47(1), pp. 5678.

Abstract: The evolution of a consumer exploiting two resources is investigated. The strategy x under selection represents the fraction of time or energy an individual invests into extracting the first resource. In the model, a dimensionless parameter alpha quantifies how simultaneous consumption of both resources influences consumer growth; alpha<0 corresponds to hemiessential resources, 0 
BibTeX:
@article{jmb03, author = {S. J. Schreiber and G. A. *Tobiason}, title = {The evolution of resource use}, journal = {J. Math. Biol.}, year = {2003}, volume = {47}, number = {1}, pages = {5678} } 
Schreiber SJ (2003), "Allee effects, chaotic transients, and unexpected extinctions", Theoretical Population Biology. Vol. 64, pp. 201209.

Abstract: Discrete time single species models with overcompensating density dependence and an Allee effect due to predator satiation and mating limitation are investigated. The models exhibit four behaviors: persistence for all initial population densities, bistability in which a population persists for intermediate initial densities and otherwise goes extinct, extinction for all initial densities, and essential extinction in which "almost every" initial density leads to extinction. For fastgrowing populations, these models show populations can persist at high levels of predatimi even though lower levels of predation lead to essential extinction. Alternatively, increasing the predator's handling time, the population's carrying capacity, or the likelihood of mating success may lead to essential extinction. In each of these cases, the mechanism behind these disappearances are chaotic dynamics driving populations below a critical threshold determined by the Allee effect. These disappearances are proceeded by chaotic transients that are proven to be approximately exponentially distributed in length and highly sensitive to initial population densities.!! 
BibTeX:
@article{tpb03, author = {S. J. Schreiber}, title = {Allee effects, chaotic transients, and unexpected extinctions}, journal = {Theoretical Population Biology}, year = {2003}, volume = {64}, pages = {201209} } 
Schreiber S, Fox L and Getz W (2002), "Parasitoid sex allocation affects coevolution of patch selection in hostparasitoid systems", Evolutionary Ecology Research. Vol. 4, pp. 701718.

Abstract: Previously, we have show that the coevolution of patch selection strategies of hosts and parasitoids in heterogeneous environments can lead to contrary habitat choices in which the hosts preferentially select patches that the parasitoids avoid. Since this work did not include the variable parasitoid sex ratios that have been found in field and laboratory systems with contrary choices, we extend previous analyses to determine how parasitoid sex allocation coevolves with host and parasitoid patch preferences. In our analysis, we assume the environment consists of two patch types: lower quality patches and higher quality patches. In the lower quality patches, hosts have a lower intrinsic rate of growth and female parasitoid larvae are less likely to survive than male parasitoid larvae. Our coevolutionary analysis reveals that the coevolved parasitoids preferentially search for hosts in higher quality patches, lay primarily female eggs on hosts encountered in these patches, and are more likely to lay male eggs on hosts encountered in the lower quality patches. As a coevolutionary response, the hosts lay twice as many eggs in the poorer patches as they would if parasitoid sex ratios did not evolve. We conclude by showing that the coevolution of parasitoid sex allocation with patch selection can stabilize hostparasitoid interactions even when coevolution of patch selection by itself does not.!! 
BibTeX:
@article{eer02, author = {S.J. Schreiber and L.R. Fox and W.M. Getz}, title = {Parasitoid sex allocation affects coevolution of patch selection in hostparasitoid systems}, journal = {Evolutionary Ecology Research}, year = {2002}, volume = {4}, pages = {701718} } 
Mierczyński J and Schreiber SJ (2002), "Kolmogorov vector fields with robustly permanent subsystems", Journal of Mathematical Analysis and Applications. Vol. 267(1), pp. 329337.

BibTeX:
@article{jmaa02, author = {Mierczyński, Janusz and S. J. Schreiber}, title = {Kolmogorov vector fields with robustly permanent subsystems}, journal = {Journal of Mathematical Analysis and Applications}, year = {2002}, volume = {267}, number = {1}, pages = {329337} } 
Schreiber SJ (2002), "Permanence of weakly coupled vector fields", SIAM J. Math. Anal.. Vol. 33(5), pp. 10481057 (electronic).

BibTeX:
@article{sima02, author = {Schreiber, Sebastian J.}, title = {Permanence of weakly coupled vector fields}, journal = {SIAM J. Math. Anal.}, year = {2002}, volume = {33}, number = {5}, pages = {10481057 (electronic)} } 
Schreiber SJ (2001), "Chaos and Sudden Extinction in Simple Ecological Models", Journal of Mathematical Biology. Vol. 42, pp. 239260.

Abstract: A class of truncated unimodal discretetime single species models for which low or high densities result in extinction in the following generation are considered. A classification of the dynamics of these maps into five types is proven: (i) extinction in finite time for all initial densities, (ii) semistability in which all orbits tend toward the origin or a semistable fixed point, (iii) bistability for which the origin and an interval bounded away from the origin are attracting, (iv) chaotic semistability in which there is an interval of chaotic dynamics whose compliment lies in the originÕs basin of attraction and (v) essential extinction in which almost every (but not every) initial population density leads to extinction in finite time. Applying these results to the Logistic, Ricker and generalized BevertonHolt maps with constant harvesting rates, two birfurcations are shown to lead to sudden population disappearances: a saddle node bifurcation corresponding to a transition from bistability to extinction and a chaotic blue sky catastrophe corresponding to a transition from bistability to essential extinction. 
BibTeX:
@article{jmb01, author = {S. J. Schreiber}, title = {Chaos and Sudden Extinction in Simple Ecological Models}, journal = {Journal of Mathematical Biology}, year = {2001}, volume = {42}, pages = {239260} } 
Schreiber S, Mills N and Gutierrez A (2001), "Hostlimited dynamics of autoparasitoids", Journal of Theoretical Biology. Vol. 212, pp. 141153.

Abstract: Autoparasitoids, an important class of intraguild predators used in classical biological control, have a unique biology. Females develop as primary endoparasitoids of scale insects and whiteflies. Males develop at the expense of conspecific or heterospecific parasitoid prepupae. To evaluate the effect of autoparasitism on host suppression, system stability, and parasitoid coexistence, stagestructured differential equation models are developed and analysed. For a hostÐparasitoid system, autoparasitism stabilizes hostÐparasitoid oscillations generated by developmental delays of the parasitoid. In hostÐautoparasitoidÐprimary parasitoid systems, a distinction between obligate (i.e. parasitoid only attacks conspecifics for the production of males) and facultative (i.e. parasitoid attacks conspecifics and heterospecifics for the production of males) autoparasitism is drawn. Coexistence between an obligate autoparasitoid and primary parasitoid occurs if and only if the autoparasitoid can invade at lower host densities than the primary parasitoid, and the primary parasitoid can suppress the host to a lower equilibrium density than the autoparasitoid. When coexistence occurs, the primary parasitoid determines the host equilibrium abundance. Interactions between facultative autoparasitoids and primary parasitoids can lead to a priority effect, and, less likely, to coexistence. When coexistence occurs, the invasion of the facultative autoparasitoid into the hostÐprimary parasitoid system raises the equilibrium density of the host. In either coexistence scenario, the invasion of an autoparasitoid can stabilize an unstable hostÐprimary parasitoid system. The analysis concludes by showing that the introduction of an autoparasitoid to a hostÐprimary parasitoid system can improve host suppression in the shortterm despite possible longterm disruption. 
BibTeX:
@article{schreiberetal01, author = {Schreiber, S.J. and Mills, N.J. and Gutierrez, A.P.}, title = {Hostlimited dynamics of autoparasitoids}, journal = {Journal of Theoretical Biology}, year = {2001}, volume = {212}, pages = {141153} } 
Schreiber SJ (2001), "Urn models, replicator processes, and random genetic drift", SIAM Journal of Applied Mathematics. Vol. 61(6), pp. 21482167 (electronic).

Abstract: To understand the relative importance of natural selection and random genetic drift in finite but growing populations, the asymptotic behavior of a class of generalized Polya urns is studied using the method of ordinary differential equation (ODE). Of particular interest is the replicator process: two balls (individuals) are chosen from an urn (the population) at random with replacement and balls of the same colors (strategies) are added or removed according to probabilities that depend only on the colors of the chosen balls. Under the assumption that the expected number of balls being added always exceeds the expected number of balls being removed whenever balls are in the urn, the probability of nonextinction is shown to be positive. On the event of nonextinction, three results are proven: (i) the number of balls increases asymptotically at a linear rate, (ii) the distribution x(n) of strategies at the nth update is a "noisy" CauchyEuler approximation to the mean limit ODE of the process, and (iii) the limit set of x(n) is almost surely a connected internally chain recurrent set for the mean limit ODE. Under a stronger set of assumptions, it is shown that for any attractor of the mean limit ODE there is a positive probability that the limit set for x(n) lies in this attractor. Theoretical and numerical estimates for the probabilities of nonextinction and convergence to an attractor suggest that random genetic drift is more likely to overcome natural selection in small populations for which pairwise interactions lead to highly variable outcomes, and is less likely to overcome natural selection in large populations with the potential for rapid growth. 
BibTeX:
@article{siap01, author = {Schreiber, S. J.}, title = {Urn models, replicator processes, and random genetic drift}, journal = {SIAM Journal of Applied Mathematics}, year = {2001}, volume = {61}, number = {6}, pages = {21482167 (electronic)} } 
Schreiber SJ, Fox LR and Getz WM (2000), "Coevolution of contrary choices in hostparasitoid systems", American Naturalist. , pp. 637648.

Abstract: We investigate patch selection strategies of hosts and parasitoids in heterogeneous environments. Previous theoretical work showed that when host traits vary among patches, coevolved populations of hosts and parasitoids make congruent choices (i.e.,hosts and parasitoids preferentially select the same patches) and exhibit direct density dependence in the distribution of percent parasitism. However, hostparasitoid systems in the field show a range of patterns in percent parasitism, while behavioral studies indicate that hosts and parasitoids can exhibit contrary choices (i.e., hosts avoid parches favored by the parasitoid). We extend previous theory by permitting lifehistory traits of the parasitoid as well as the host to vary among patches. Our analysis implies that in coevolutionarily stable populations, hosts preferentially select patches that intrinsically support higher host equilibrium numbers (i.e., the equilibrium number achieved by hosts when both populations are confined toa single patch) and that parasitoids preferentially select patches that intrinsically support higher parasitoid equilibrium numbers (i.e., the equilibrium number achieved by the parasitoids when both populations are confined to a patch). Using this result, we show how variation in lifehistory traits among patches leads to contrary or congruent choices or leads to direct density dependence, inverse density dependence, or density independence in the distribution of percent parasitism. In addition, we determine when populations playing the coevolutionarily stable strategies are ecologically stable. Our analysis shows that heterogeneous environments containing patches where the intrinsic rate of growth of the host and the survivorship rate ofthe parasitoid are low result in the coevolved populations exhibiting contrary choices and, as a result, promote ecological stability. 
BibTeX:
@article{amnat00, author = {S. J. Schreiber and L. R. Fox and W. M. Getz}, title = {Coevolution of contrary choices in hostparasitoid systems}, journal = {American Naturalist}, year = {2000}, pages = {637648} } 
Eisenberg JNS, Washburn JO and Schreiber SJ (2000), "The generalist feeding behaviors of Aedes sierrensis larvae and their effects on protozoan populations", Ecology. Vol. 81, pp. 921935.

Abstract: The generalist feeding strategy of larvae of the western tree hole mosquito, Aedes sierrensis, is central to understanding the communitylevel effects of the tritrophic interactions among mosquito larvae, midsized organisms (such as protozoa), and lowerlevel organisms (such as bacteria and fungi) in west coast phytotelmata. Laboratory microcosm experiments were conducted to characterize the feeding strategies of Ae. sierrensis larvae in the presence of multiple resource types (freeswimming protozoa and substratebound particulate material). In our experiment, we quantified the effects of varying instar numbers and profile, resource type, and refuge size on predation of protozoa. Refugia were explicitly modeled in our microcosms, representing the interstitial spaces of leaf litter and the wood lining of natural tree holes. Results from these microcosm experiments suggested that: (1) Even in the absence of larvae, the majority of protozoa resided in the smallvolume, resourcerich refugia. There was, however, a strong nonlinear and negative relationship between larval densities in the upper compartment and the protozoan densities in the refuge, suggesting that there was continual movement of protozoa between the two spaces. (2) Fourth instars harvested resources by filterfeeding at a higher rate than second instars. (3) As the level of substratebound particulate food was increased, the predation pressure by filterfeeding on the protozoa decreased. (4) As the refuge volume increased, the predation pressure on the protozoa decreased. We constructed a threestatevariable mathematical model describing the generalist feeding behavior of Ae. sierrensis larvae. The model system, with constant predator densities and two prey groups, exhibited full cooperativity; i.e., an increase in protozoa density resulted in a shift toward predation by filter feeding, while an increase in substratebound resources resulted in a shift toward predation by browsing. This indirect mutualism is mechanistically distinct from previously published systems and provides a potential mechanism for protozoan persistence in the presence of larval predation. 
BibTeX:
@article{ecology00, author = {J. N. S. Eisenberg and J. O. Washburn and S. J. Schreiber}, title = {The generalist feeding behaviors of Aedes sierrensis larvae and their effects on protozoan populations}, journal = {Ecology}, year = {2000}, volume = {81}, pages = {921935} } 
Benaïm M and Schreiber SJ (2000), "Ergodic properties of weak asymptotic pseudotrajectores for semiflows", Journal of Dynamics and Differential Equations. Vol. 12, pp. 579598.

BibTeX:
@article{jdde00, author = {M. Benaïm and S. J. Schreiber}, title = {Ergodic properties of weak asymptotic pseudotrajectores for semiflows}, journal = {Journal of Dynamics and Differential Equations}, year = {2000}, volume = {12}, pages = {579598} } 
Schreiber SJ (2000), "Criteria for $C^r$ robust permanence", Journal of Differential Equations. , pp. 400426.

Abstract: Let $dot x_i=x_if_i(x)$ ($i=1,n$) be a $C^r$ vector field that generates a dissipative flow $ on the positive cone of $. $ is called permanent if the boundary of the positive cone is repelling. $ is called br>$C^r$ robustly permanent if $ remains permanent for sufficiently small $C^r$ perturbations of the vector field. A necessary condition and a sufficient condition for $C^r$ robust permanence involving the average percapita growth rates $int f_i d with respect to invariant measures $ are derived. The necessary condition requires that $mu i int f_i d0$ where the infimum is taken over ergodic measures with compact support in the boundary of the positive cone. The sufficient condition requires that the boundary flow admits a Morse decomposition $M_1, M_k$ such that every $M_j$ satisfies $mu i int f_i dmu >0$ where the minimum is taken over invariant measures with support in $M_j$. As applications, we provide a sufficient condition for $C^r$ robust permanence of LotkaVolterra models and a topological characterization of $C^r$ robust permanence for food chain models. !! 
BibTeX:
@article{jde00, author = {S. J. Schreiber}, title = {Criteria for $C^r$ robust permanence}, journal = {Journal of Differential Equations}, year = {2000}, pages = {400426} } 
Getz WM and Schreiber SJ (1999), "Multiple time scales in consumerresource interactions", Annales Zooligici Fennici. Vol. 36, pp. 1120.

Abstract: Arguments regarding the appropriate form for the rate at which consumers extract biomass from resource populations hinge on relative time scales of underlying processes. Some ecologists argue that, because differential equation models imply instantaneous rates of change, time scale arguments do not hold. Here we point out that this reasoning is fallacious. We define three natural time scales for consumerresource interactions and demonstrate, using asymptotic methods of analysis, how relative differences in these scales lead to the formulation of models with qualitatively distinct dynamics. Further, we identify time scale and resource heterogeneity assumptions that constrain the R~* competition rule (i.e., the competitor that suppresses theresource to the lowest density excludes all other competitors), as well as clarify the dichotomy between Schoener's models of competition for overlapping and for partitioned resources. 
BibTeX:
@article{azf99, author = {W. M. Getz and S. J. Schreiber}, title = {Multiple time scales in consumerresource interactions}, journal = {Annales Zooligici Fennici}, year = {1999}, volume = {36}, pages = {1120} } 
Schreiber SJ and Gutierrez AP (1999), "Insect invasions and community assembly", In Ecological Entomology. New York , pp. 425462. John Wiley & Sons.

BibTeX:
@inproceedings{book99, author = {S. J. Schreiber and A. P. Gutierrez}, editor = {A. P. Gutierrez and C. B. Huffaker}, title = {Insect invasions and community assembly}, booktitle = {Ecological Entomology}, publisher = {John Wiley & Sons}, year = {1999}, pages = {425462} } 
Schreiber SJ (1999), "Successional stability of vector fields in dimension three", Proceedings of the American Mathematical Soceity. Vol. 127, pp. 9931002.

BibTeX:
@article{pams99, author = {S. J. Schreiber}, title = {Successional stability of vector fields in dimension three}, journal = {Proceedings of the American Mathematical Soceity}, year = {1999}, volume = {127}, pages = {9931002} } 
Schreiber SJ (1998), "On the stabilizing effect of specialist predators on founder controlled communities", Canadian Applied Mathematical Quarterly. Vol. 6, pp. 112.

Abstract: We study a generalized model of 2n interacting species consisting of n competing prey and n predators, each of which feeds exclusively upon one of the prey species. Under the assumption that the prey community is foundercontrolled (the positive equilibria determined by single prey species are asymptotically stable in prey phase space), it is shown that then predators can mediate uniform persistence when their mortality rates are sufficiently small. When this occurs, a repelling heteroclinic network on the boundary of the positive orthant is formed in which the removal of any predator leads to a system with a globally asymptotically stable equilibrium that only supports a single species.!! 
BibTeX:
@article{camq98, author = {S. J. Schreiber}, title = {On the stabilizing effect of specialist predators on founder controlled communities}, journal = {Canadian Applied Mathematical Quarterly}, year = {1998}, volume = {6}, pages = {112} } 
Regev U, Gutierrez AP, Schreiber SJ and Zilberman D (1998), "Bioeconomic foundations of renewable resource exploitation", Ecological Economics. Vol. 26, pp. 227242.

BibTeX:
@article{ecoecon98, author = {U. Regev and A. P. Gutierrez and S. J. Schreiber and D. Zilberman}, title = {Bioeconomic foundations of renewable resource exploitation}, journal = {Ecological Economics}, year = {1998}, volume = {26}, pages = {227242} } 
Schreiber SJ and Gutierrez AP (1998), "A supplydemand perspective of species invasions: Applications to biological control", Ecological Modelling. Vol. 106, pp. 2745.

BibTeX:
@article{ecomod98, author = {S. J. Schreiber and A. P. Gutierrez}, title = {A supplydemand perspective of species invasions: Applications to biological control}, journal = {Ecological Modelling}, year = {1998}, volume = {106}, pages = {2745} } 
Schreiber SJ (1998), "On growth rates of subadditive functions for semiflows", Journal of Differential Equations. Vol. 148, pp. 334350.

BibTeX:
@article{jde98, author = {S. J. Schreiber}, title = {On growth rates of subadditive functions for semiflows}, journal = {Journal of Differential Equations}, year = {1998}, volume = {148}, pages = {334350} } 
Schreiber SJ (1997), "Expansion rates and Lyapunov exponents", Discrete and Continous Dynamical Systems. Vol. 3, pp. 433438.

BibTeX:
@article{dcds97, author = {S. J. Schreiber}, title = {Expansion rates and Lyapunov exponents}, journal = {Discrete and Continous Dynamical Systems}, year = {1997}, volume = {3}, pages = {433438} } 
Schreiber SJ (1997), "Gerneralist and specialist predators that mediate permanence in ecological communities", Journal of Mathematical Biology. Vol. 36, pp. 133148.

Abstract: General dynamic models of systems with two prey and one or two predators are considered. After rescaling the equations so that both prey have the same intrinsic rate of growth, it is shown that there exists a generalist predator that can mediate permanence if and only if there is a population density of a prey at which its percapita growth rate is positive yet less than its competitorÕs invasion rate. In particular, this result implies that if the outcome of competition between the prey is independent of initial conditions, then there exists a generalist predator that mediates permanence. On the other hand, if the outcome of competition is contingent upon initial conditions (i.e., the prey are bistable), then there may not exist a suitable generalist predator. For example, bistable prey modeled by the AyalaÐGilpin (thetaLogistic) equations can be stabilized if and only if theta<1 for one of the prey. It is also shown that two specialist predators always can mediate permanence between bistable prey by creating a repelling heteroclinic cycle consisting of fixed points and limit cycles. 
BibTeX:
@article{jmb97, author = {S. J. Schreiber}, title = {Gerneralist and specialist predators that mediate permanence in ecological communities}, journal = {Journal of Mathematical Biology}, year = {1997}, volume = {36}, pages = {133148} } 
Schreiber SJ (1996), "Global stability in consumerresource cascades", Journal of Mathematical Biology. Vol. 35, pp. 3748.

Abstract: Models of population growth in consumerresource cascades (serially arranged containers with a dynamic consumer population, v, receiving a flow of resource, u, from the previous container) with a functional response of the form h(u/v^b) are investigated. For 1>b>0, it is shown that these models have a globally stable equilibrium. As a result, two conclusions can be drawn: (1) Consumer density dependence in the functional or in the percapita numerical response can result in persistence of the consumer population in all containers. (2) In the absence of consumer density dependence, the consumer goes extinct in all containers except possibly the first. Several variations of this model are discussed including replacing discrete containers by a spatial continuum and introducing a dynamic resource. 
BibTeX:
@article{jmb96, author = {S. J. Schreiber}, title = {Global stability in consumerresource cascades}, journal = {Journal of Mathematical Biology}, year = {1996}, volume = {35}, pages = {3748} } 
Schreiber SJ (1995), "Nonuniformly hyperbolic dynamics". Thesis at: University of California, Berkeley.

BibTeX:
@phdthesis{schreiber95, author = {S. J. Schreiber}, title = {Nonuniformly hyperbolic dynamics}, school = {University of California, Berkeley}, year = {1995} } 
Gutierrez AP, Mills NJ, Schreiber SJ and Ellis CK (1994), "A phsysiologically based tritrophic perspective on bottomup topdown regulation of populations", Ecology. Vol. 75, pp. 22272242.

BibTeX:
@article{ecology94, author = {A. P. Gutierrez and N. J. Mills and S. J. Schreiber and C. K. Ellis}, title = {A phsysiologically based tritrophic perspective on bottomup topdown regulation of populations}, journal = {Ecology}, year = {1994}, volume = {75}, pages = {22272242} } 