Robust Permanence for Ecological Differential Equations, Minimax, and Discretizations

Garay, Barnabás M.; Hofbauer, Josef

doi:10.1137/s0036141001392815

Cited by 68 publications

(100 citation statements)

References 54 publications

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“…They showed that for certain systems persistence holds even when one has small perturbations of the growth functions. There have been results on robust persistence in the deterministic setting for Kolmogorov systems by Schreiber (2000) and Garay and Hofbauer (2003). Recently, robust permanence for deterministic Kolmogorov equations with respect to perturbations in both the growth functions and the feedback dynamics has been analyzed by Patel and Schreiber (2016).…”

Section: Discussion and Generalizationsmentioning

confidence: 99%

Symmetries and pattern formation in hyperbolic versus parabolic models of self-organised aggregation

Buono

Eftimie

2014

J. Math. Biol.

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This work is devoted to studying the dynamics of a structured population that is subject to the combined effects of environmental stochasticity, competition for resources, spatio-temporal heterogeneity and dispersal. The population is spread throughout n patches whose population abundances are modeled as the solutions of a system of nonlinear stochastic differential equations living on [0, ∞) n . We prove that r , the stochastic growth rate of the total population in the absence of competition, determines the long-term behaviour of the population. The parameter r can be expressed as the Lyapunov exponent of an associated linearized system of stochastic differential equations. Detailed analysis shows that if r > 0, the population abundances converge polynomially fast to a unique invariant probability measure on (0, ∞) n , while when r < 0, the population abundances of the patches converge almost surely to 0 exponentially fast. 2013) and proves one of their conjectures. Compared to recent developments, our model incorporates very general density-dependent growth rates and competition terms. Furthermore, we prove that persistence is robust to small, possibly density dependent, perturbations of the growth rates, dispersal matrix and covariance matrix of the environmental noise. We also show that the stochastic growth rate depends continuously on the coefficients. Our work allows the environmental noise driving our system to be degenerate. This is relevant from a biological point of view since, for example, the environments of the different patches can be perfectly correlated. We show how one can adapt the nondegenerate results to the degenerate setting.As an example we fully analyze the two-patch case, n = 2, and show that the stochastic growth rate is a decreasing function of the dispersion rate. In particular, coupling two sink patches can never yield persistence, in contrast to the results from the nondegenerate setting treated by Evans et al. which show that sometimes coupling by dispersal can make the system persistent.

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Section: Discussion and Generalizationsmentioning

confidence: 99%

Symmetries and pattern formation in hyperbolic versus parabolic models of self-organised aggregation

Buono

Eftimie

2014

J. Math. Biol.

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“…The papers by Faria and Röst [12], Freedman and Ruan [15], Garay and Hofbauer [16], Hetzer and Shen [20], Hirsch et al [21], Langa et al [26], Magal and Zhao [28], Mierczyński and Shen [30], Mierczyński et al [31], Novo et al [36], Salceanu and Smith [43], Schreiber [44,45], Thieme [51,52], Wang and Zhao [54], and references therein, provide a long but not complete list of works on this topic.…”

Section: Introductionmentioning

confidence: 99%

Uniform and strict persistence in monotone skew-product semiflows with applications to non-autonomous Nicholson systems

Obaya

Sanz

2016

Journal of Differential Equations

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Abstract. We determine sufficient conditions for uniform and strict persistence in the case of skew-product semiflows generated by solutions of nonautonomous families of cooperative systems of ODEs or delay FDEs in terms of the principal spectrums of some associated linear skew-product semiflows which admit a continuous separation. Our conditions are also necessary in the linear case. We apply our results to a noncooperative almost periodic Nicholson system with a patch structure, whose persistence turns out to be equivalent to the persistence of the linearized system along the null solution.

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“…Mathematical modeling is especially useful for exploring these idealized scenarios. Furthermore, permanence, or persistence in mathematical models is known to be robust to model perturbations under appropriate conditions [10,3,5] and therefore it should continue to hold for small deviations from a nested infection structure.…”

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confidence: 99%

How nested and monogamous infection networks in host-phage communities come to be

Korytowski

Smith

2014

Theor Ecol

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We show that a chemostat community of bacteria and bacteriophage in which bacteria compete for a single nutrient and for which the bipartite infection network is perfectly nested is permanent, a.k.a. uniformly persistent, provided that bacteria that are superior competitors for nutrient devote the least effort to defence against infection and the virus that are the most efficient at infecting host have the smallest host range. This confirms earlier work of Jover et al [7] who raised the issue of whether nested infection networks are permanent. In addition, we provide sufficient conditions that a bacteria-phage community of arbitrary size with nested infection network can arise through a succession of permanent subcommunties each with a nested infection network by the successive addition of one new population. The same permanence results hold for the monogamous infection network considered by Thingstad [15] but without the tradeoffs.

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Robust Permanence for Ecological Differential Equations, Minimax, and Discretizations

Cited by 68 publications

References 54 publications

Symmetries and pattern formation in hyperbolic versus parabolic models of self-organised aggregation

Symmetries and pattern formation in hyperbolic versus parabolic models of self-organised aggregation

Uniform and strict persistence in monotone skew-product semiflows with applications to non-autonomous Nicholson systems

How nested and monogamous infection networks in host-phage communities come to be

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