Chaotic dynamics in a simple predator-prey model with discrete delay

A discrete delay is included to model the time between the capture of the prey and its conversion to viable biomass in the simplest classical Gause type predator-prey model that has equilibrium dynamics without delay. As the delay increases from zero, the coexistence equilibrium undergoes a supercritical Hopf bifurcation, two saddle-node bifurcations of limit cycles, and a cascade of period doublings, eventu1ally leading to chaos. The resulting periodic orbits and the strange attractor resemble their counterparts for the Mackey-Glass equation. Due to the global stability of the system without delay, these complicated dynamics can be solely attributed to the introduction of the delay. Since many models include predator-prey like interactions as submodels, this study emphasizes the importance of understanding the implications of overlooking delay in such models on the reliability of the model-based predictions, especially since the temperature is known to have an effect on the length of certain delays.Comment: This paper has 28 pages, 12 figures and has been accepted to DCDS-B. Please cite the journal version once it is published in DCDS-B. Appreciate tha

Population Growth and Competition Models with Decay and Competition Consistent Delay

We derive an alternative expression for a delayed logistic equation in which the rate of change in the population involves a growth rate that depends on the population density during an earlier time period. In our formulation, the delay in the growth term is consistent with the rate of instantaneous decline in the population given by the model. Our formulation is a modification of [Arino et al., J.~Theoret.~Biol.~241(1):109--119, 2006] by taking the intraspecific competition between the adults and juveniles into account. We provide a complete global analysis showing that no sustained oscillations are possible. A threshold giving the interface between extinction and survival is determined in terms of the parameters in the model. The theory of chain transitive sets and the comparison theorem for cooperative delay differential equations are used to determine the global dynamics of the model. We extend our delayed logistic equation to a system modeling the competition of two species. For the competition model, we provide results on local stability, bifurcation diagrams, and adaptive dynamics. Assuming that the species with shorter delay produces fewer offspring at a time than the species with longer delay, we show that there is a critical value, $\tau^*$, such that the evolutionary trend is for the delay to approach $\tau^*$.Comment: 22 pages, 6 figures, 1 tabl

A Predator-Prey Model in the Chemostat with Time Delay

The aim of this paper is to study the dynamics of predator-prey interaction in a chemostat to determine whether including a discrete delay to model the time between the capture of the prey and its conversion to viable biomass can introduce oscillatory dynamics even though there is a globally asymptotically stable equilibrium when the delay is ignored. Hence, Holling type I response functions are chosen so that no oscillatory behavior is possible when there is no delay. It is proven that unlike the analogous model for competition, as the parameter modeling the delay is increased, Hopf bifurcations can occur

Global asymptotic behavior of a multi-species stochastic chemostat model with discrete delays

We consider a model of multi-species competition in the chemostat that includes demographic stochasticity and discrete delays. We prove that for any given initial data, there exists a unique global positive solution for the stochastic delayed system. By employing the method of stochastic Lyapunov functionals, we determine the asymptotic behaviors of the stochastic solution and show that although the sample path fluctuate, it remains positive and the expected time average of the distance between the stochastic solution and the equilibrium of the associated deterministic delayed chemostat model is eventually small, i.e. we obtain an analogue of the competition exclusion principle when the noise intensities are relatively small. Numerical simulations are carried out to illustrate our theoretical results.Accepted versio

Competition In The Chemostat: A Distributed Delay Model And Its Global Asymptotic Behavior

. In this paper, we propose a two species competition model in a chemostat that uses a distributed delay to model the lag in the process of nutrient conversion and study the global asymptotic behavior of the model. The model includes a washout factor over the time delay involved in the nutrient conversion and hence the delay is distributed over the species concentrations as well as over the nutrient concentration (using the gamma distribution). The results are valid for a very general class of monotone growth response functions. By using the linear chain trick technique and the Fluctuation Lemma, we completely determine the global limiting behavior of the model and prove that there is always at most one survivor and give a criterion to predict the outcome that is dependent upon the parameters in the delay kernel. We compare these predictions on the qualitative outcome of competition introduced by including distributed delay in the model with the predictions made by the the correspondin..