Robust permanence for ecological equations with internal and external feedbacks

Robust permanence for ecological maps

We consider ecological difference equations of the form $X_{t+1}^i =X_t^i A_i(X_t)$ where $X_t^i$ is a vector of densities corresponding to the subpopulations of species $i$ (e.g. subpopulations of different ages or living in different patches), $X_t=(X_t^1,X_t^2,\dots,X_t^m)$ is state of the entire community, and $A_i(X_t)$ are matrices determining the update rule for species $i$. These equations are permanent if they are dissipative and the extinction set $\{X: \prod_i \|X^i\|=0\}$ is repelling. If permanence persists under perturbations of the matrices $A_i(X)$, the equations are robustly p…

Evolutionarily induced alternative states and coexistence in systems with apparent competition

Predators often consume multiple prey and by mutually subsidizing a shared predator, the prey may reciprocally harm each other. When predation levels are high, this apparent competition can culminate in a prey species being displaced. Coupling quantitative genetics and Lotka-Volterra models, we study how predator evolution alters this and other ecological outcomes. These models account for a trade-off between the predator's attack rates on two prey species. We provide a mathematical characterization of a strong form of persistence--permanence--for which there is a global attractor bounded away…

具有非单调人均增长率的一般离散时间两种群相互作用模型的持久性

Permanence of a general discrete-time two-species-interaction model with non-monotonic per capita growth rates

结合所有依赖密度的因素，物种的人均增长率可能是非单调的。一个重要的结果是，物种可能遭受弱Allee效应或强Allee效应。本文首次研究了具有非单调人均增长率的离散时间两种群相互作用模型的持久性。利用平均李雅普诺夫函数，推广了相对非线性的生态学概念，得到了一个简单的保证系统持久生存的充分条件.扩展的相对非线性使我们能够充分表征非单调性的人均增长函数的非线性的影响。这些结果说明了具体的两个物种的竞争和捕食者-食饵模型的通用形式与非单调的人均增长率。

n-物种Lotka-Volterra模型在周期脉冲下的持久性

Persistence of n-Species Lotka-Volterra Models with Periodic Pulses

针对周期脉冲干扰下的n-物种Lotka-Volterra模型，提出基于Morse分解和加权长期增长率的持久性充分条件，并证明其鲁棒性，应用于癌症化疗与农业害虫防治案例。

Permanence via invasion graphs: Incorporating community assembly into Modern Coexistence Theory

To understand the mechanisms underlying species coexistence, ecologists often study invasion growth rates of theoretical and data-driven models. These growth rates correspond to average per-capita growth rates of one species with respect to an ergodic measure supporting other species. In the ecological literature, coexistence often is equated with the invasion growth rates being positive. Intuitively, positive invasion growth rates ensure that species recover from being rare. To provide a mathematically rigorous framework for this approach, we prove theorems that answer two questions: (i) When…

Persistence and extinction for stochastic ecological models with internal and external variables

The dynamics of species' densities depend both on internal and external variables. Internal variables include frequencies of individuals exhibiting different phenotypes or living in different spatial locations. External variables include abiotic factors or non-focal species. These internal or external variables may fluctuate due to stochastic fluctuations in environmental conditions. We prove theorems for stochastic persistence and exclusion for stochastic ecological difference equations accounting for internal and external variables. Specifically, we use a stochastic analog of average Lyapuno…