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

Hopf Bifurcation in a Modified Leslie-Gower Two Preys One Predator Model and Holling Type II Functional Response with Harvesting and Time-Delay

In this paper, a modified Leslie-Gower two preys one predator model and Holling type II functional response with harvesting and time-delay were discussed. Model analysis is carried out by determining fixed points, then analyzing the stability of the fixed points and discussing the existence of the Hopf bifurcation. In some conditions that occur in nature indicate the occurrence of hunting of prey and predator species by humans. Therefore, this model is modified by adding the assumption that prey and predators are being harvested. Another modification given to the model is the use of time delays.The delay time term is for taking into account the case that the members of the predator species need time from birth to predation for being active predators. The first case is a model without time delay, it is obtained that 3 fixed points are unstable and 7 fixed points are stable. One of them is the interior fixed point tested with the Routh-Hurwitz criteria. The second case is a model with a delay time, the critical delay value is obained. Hopf bifurcation occurs when the delay time value is equal to the critical delay value and also fulfills the transversality condition. Observations on the model simulation are carried out by varying the value of the delay time. When the Hopf bifurcation occurs, the graph on the solution plane shows a constant oscillatory movement. If the value of the delay time given is less than the critical value of the delay, the controlled system solution goes to a balanced state. Then when the delay time value is greater than the critical delay value, the system solution continues to fluctuate causing an unstable system condition

Stability and Bifurcation Analysis of Time Delayed Prey-Predator System with Holling Type-III Response Function

Interaction between prey and predator is a recurring event that occurs continuously and the presence of both can mutually affect each other’s population. This paper discusses the stability and bifurcation analysis of time delayed prey-predator system with Holling type-III response function incorporating a prey refuge. Holling type-III response function is used because the availability of the prey in nature is decreasing. Time delay represents the time for predators to find another prey population when the available population is decreasing. Dynamic analysis is used to study the influence of a given response function. The equilibrium point and stability analysis of the model with and without time delay has been calculated and analyzed. Model analysis under the influence of time delay and coefficient of competition among predators shows an indication of Hopf bifurcation in the neighborhood of the co-existing equilibrium point