Related papers: Robust permanence for ecological maps
The dynamics of interacting structured populations can be modeled by $\frac{dx_i}{dt}= A_i (x)x_i$ where $x_i\in \R^{n_i}$, $x=(x_1,\dots,x_k)$, and $A_i(x)$ are matrices with non-negative off-diagonal entries. These models are permanent if…
Species experience both internal feedbacks with endogenous factors such as trait evolution and external feedbacks with exogenous factors such as weather. These feedbacks can play an important role in determining whether populations persist…
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…
Let $(X_t)_{t \geq 0}$ be a continuous time Markov process on some metric space $M,$ leaving invariant a closed subset $M_0 \subset M,$ called the {\em extinction set}. We give general conditions ensuring either "Stochastic persistence"…
This paper provides a necessary and sufficient condition for guaranteeing exponential stability of the linear difference equation $x(t)=Ax(t-a)+Bx(t-b)$ where $a>0,b>0$ are constants and $A,B$ are $n\times n$ square matrices, in terms of a…
We show how highly-diverse ecological communities may display persistent abundance fluctuations, when interacting through resource competition and subjected to migration from a species pool. This turns out to be closely related to the ratio…
Individuals within any species exhibit differences in size, developmental state, or spatial location. These differences coupled with environmental fluctuations in demographic rates can have subtle effects on population persistence and…
We analyze ecological systems that are influenced by random environmental fluctuations. We first provide general conditions which ensure that the species coexist and the system converges to a unique invariant probability measure (stationary…
We consider a system of nonlinear partial differential equations that describes an age-structured population inhabiting several temporally varying patches. We prove existence and uniqueness of solution and analyze its large-time behavior in…
We study the stability of $\mathcal{M}_0$, an invariant subset of a Markov process $(X_t)_{t\geq 0}$ on a metric space $\mathcal{M}$. By building the theory of average Lyapunov functions, we formulate general criteria based on the signs of…
This paper investigates the dynamics of vegetation patterns in water-limited ecosystems using a generalized Klausmeier model that incorporates non-local plant dispersal within a finite habitat. We establish the well-posedness of the system…
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…
Over the past century, nonlinear difference and differential equations have been used to understand conditions for species coexistence. However, these models fail to account for random fluctuations due to demographic and environmental…
This paper is devoted to the study of the persistence versus extinction of species in the reaction-diffusion equation: \begin{equation} u_t-\Delta u=f(t,x_1-ct,y,u) \quad\quad t>0,\ x\in\Omega,\nonumber \end{equation} where $\Omega$ is of…
A persistent dynamical system in $\mathbb{R}^d_{> 0}$ is one whose solutions have positive lower bounds for large $t$, while a permanent dynamical system in $\mathbb{R}^d_{> 0}$ is one whose solutions have uniform upper and lower bounds for…
We study the biodiversity problem for resource competition systems with extinctions and self-limitation effects. Our main result establishes estimates of biodiversity in terms of the fundamental parameters of the model. We also prove the…
This paper investigates the deterministic extinction and permanence of a family of SEIRS malaria models with multiple random delays, and with a general nonlinear incidence rate. The conditions for the extinction and permanence of the…
A general system of difference equations is presented for multispecies communities with density dependent population growth and delayed maturity. Interspecific competition, mutualism, predation, commensalism, and amensalism are…
The local persistence R(t), defined as the proportion of the system still in its initial state at time t, is measured for the Bak--Sneppen model. For 1 and 2 dimensions, it is found that the decay of R(t) depends on one of two classes of…
We consider a system of nonlinear partial differential equations that describes an age-structured population living in changing environment on $N$ patches. We prove existence and uniqueness of solution and analyze large time behavior of the…