Noise in ecosystems: a short review

Noise, through its interaction with the nonlinearity of the living systems, can give rise to counter-intuitive phenomena such as stochastic resonance, noise-delayed extinction, temporal oscillations, and spatial patterns. In this paper we briefly review the noise-induced effects in three different ecosystems: (i) two competing species; (ii) three interacting species, one predator and two preys, and (iii) N-interacting species. The transient dynamics of these ecosystems are analyzed through generalized Lotka-Volterra equations in the presence of multiplicative noise, which models the interaction between the species and the environment. The interaction parameter between the species is random in cases (i) and (iii), and a periodical function, which accounts for the environmental temperature, in case (ii). We find noise-induced phenomena such as quasi-deterministic oscillations, stochastic resonance, noise-delayed extinction, and noise-induced pattern formation with nonmonotonic behaviors of patterns areas and of the density correlation as a function of the multiplicative noise intensity. The asymptotic behavior of the time average of the \emph{$i^{th}$} population when the ecosystem is composed of a great number of interacting species is obtained and the effect of the noise on the asymptotic probability distributions of the populations is discussed.Comment: 27 pages, 16 figures. Accepted for publication in Mathematical Biosciences and Engineerin

Bifurcation in a Discrete Competition System

A new difference system is induced from a differential competition system by different discrete methods. We give theoretical analysis for local bifurcation of the fixed points and derive the conditions under which the local bifurcations such as flip occur at the fixed points. Furthermore, one- and two-dimensional diffusion systems are given when diffusion terms are added. We provide the Turing instability conditions by linearization method and inner product technique for the diffusion system with periodic boundary conditions. A series of numerical simulations are performed that not only verify the theoretical analysis, but also display some interesting dynamics

Predator Extinction arose from Chaos of the Prey: the Chaotic Behavior of a Homomorphic Two-Dimensional Logistic Map in the Form of Lotka-Volterra Equations

A two-dimensional homomorphic logistic map that preserves features of the Lotka-Volterra equations was proposed. To examine chaos, iteration plots of the population, Lyapunov exponents calculated from Jacobian eigenvalues of the $2$D logistic mapping, and from time series algorithms of Rosenstein and Eckmann et al. were calculated. Bifurcation diagrams may be divided into four categories depending on topological shapes. Our model not only recovered the $1$D logistic map, which exhibits flip bifurcation, for the prey when there is a nonzero initial predator population, but it can also simulate normal competition between two species with equal initial populations. Despite the possibility for two species to go into chaos simultaneously, where the Neimark-Sacker bifurcation was observed, it is also possible that with the same interspecies parameters as normal but with a predator population $10$ times more than that of the prey, the latter becomes chaotic, while the former dramatically reduces to zero with only a few iterations, indicating total annihilation of the predator species. Interpreting humans as predators and natural resources as preys in the ecological system, the above-mentioned conclusion may imply that not only excessive consumption of natural resources, but its chaotic state triggered by an overpopulation of humans may backfire in a manner of total extinction of the human species. Fortunately, there is little chance for the survival of the human race, as isolated fixed points in the bifurcation diagram of the predator reveal. Finally, two possible applications of the phenomenon of chaotic extinction are proposed: one is to inhibit viruses or pests by initiating the chaotic states of the prey on which the viruses or pests rely for existence, and the other is to achieve the superconducting state with the chaotic state of the applied magnetic field.Comment: Paper abstract presented at the 33rd Annual Conference of the Society for Chaos Theory in Psychology & Life Sciences, The Fields Institute, U of Toronto, Aug. 4, 202

Multi-parameter reaction–diffusion systems with quadratic nonlinearity and delays: new exact solutions in elementary functions

The study considers a nonlinear multi-parameter reaction–diffusion system of two Lotka–Volterra-type equations with several delays. It treats both cases of different diffusion coefficients and identical diffusion coefficients. The study describes a few different techniques to solve the system of interest, including (i) reduction to a single second-order linear ODE without delay, (ii) reduction to a system of three second-order ODEs without delay, (iii) reduction to a system of three first-order ODEs with delay, (iv) reduction to a system of two second-order ODEs without delay and a linear Schrödinger-type PDE, and (v) reduction to a system of two first-order ODEs with delay and a linear heat-type PDE. The study presents many new exact solutions to a Lotka–Volterra-type reaction–diffusion system with several arbitrary delay times, including over 50 solutions in terms of elementary functions. All of these are generalized or incomplete separable solutions that involve several free parameters (constants of integration). A special case is studied where a solution contains infinitely many free parameters. Along with that, some new exact solutions are obtained for a simpler nonlinear reaction–diffusion system of PDEs without delays that represents a special case of the original multi-parameter delay system. Several generalizations to systems with variable coefficients, systems with more complex nonlinearities, and hyperbolic type systems with delay are discussed. The solutions obtained can be used to model delay processes in biology, ecology, biochemistry and medicine and test approximate analytical and numerical methods for reaction–diffusion and other nonlinear PDEs with delays

Existence and instability of steady states for a triangular cross-diffusion system: a computer-assisted proof

In this paper, we present and apply a computer-assisted method to study steady states of a triangular cross-diffusion system. Our approach consist in an a posteriori validation procedure, that is based on using a fxed point argument around a numerically computed solution, in the spirit of the Newton-Kantorovich theorem. It allows us to prove the existence of various non homogeneous steady states for different parameter values. In some situations, we get as many as 13 coexisting steady states. We also apply the a posteriori validation procedure to study the linear stability of the obtained steady states, proving that many of them are in fact unstable

Mathematical Analysis for a Discrete Predator-Prey Model with Time Delay and Holling II Functional Response

This paper is concerned with a discrete predator-prey model with Holling II functional response and delays. Applying Gaines and Mawhin's continuation theorem of coincidence degree theory and the method of Lyapunov function, we obtain some sufficient conditions for the existence global asymptotic stability of positive periodic solutions of the model