Switching model with two habitats and a predator involving group defence

Switching model with one predator and two prey species is considered. The prey species have the ability of group defence. Therefore, the predator will be attracted towards that habitat where prey are less in number. The stability analysis is carried out for two equilibrium values. The theoretical results are compared with the numerical results for a set of values. The Hopf bifuracation analysis is done to support the stability results

Omnivory by planktivores stabilizes plankton dynamics, but may either promote or reduce algal biomass

Classical models of phytoplankton–zooplankton interaction show that with nutrient enrichment such systems may abruptly shift from limit cycles to stable phytoplankton domination due to zooplankton predation by planktivorous fish. Such models assume that planktivorous fish eat only zooplankton, but there are various species of filter-feeding fish that may also feed on phytoplankton. Here, we extend these classical models to systematically explore the effects of omnivory by planktivorous fish. Our analysis indicates that if fish forage on phytoplankton in addition to zooplankton, the alternative attractors predicted by the classical models disappear for all realistic parameter settings, even if omnivorous fish have a strong preference for zooplankton. Our model also shows that the level of fish biomass above which zooplankton collapse should be higher when fish are omnivorous than when fish are zooplanktivorous. We also used the model to explore the potential effects of the now increasingly common practice of stocking lakes with filter-feeding fish to control cyanobacteria. Because omnivorous filter-feeding fish forage on phytoplankton as well as on the main grazers of phytoplankton, the net effect of such fish on the phytoplankton biomass is not obvious. Our model suggests that there may be a unimodal relationship between the biomass of omnivorous filter-feeding fish and the biomass of phytoplankton. This implies that to manage for reductions in phytoplankton biomass, heavy stocking or strong reduction of such fish is bes

Theoretical Study of Pest Control Using Stage Structured Natural Enemies with Maturation Delay: A Crop-Pest-Natural Enemy Model

In the natural world, there are many insect species whose individual members have a life history that takes them through two stages, immature and mature. Moreover, the rates of survival, development, and reproduction almost always depend on age, size, or development stage. Keeping this in mind, in this paper, a three species crop-pest-natural enemy food chain model with two stages for natural enemies is investigated. Using characteristic equations, a set of sufficient conditions for local asymptotic stability of all the feasible equilibria is obtained. Moreover, using approach as in (Beretta and Kuang, 2002), the possibility of the existence of a Hopf bifurcation for the interior equilibrium with respect to maturation delay is explored, which shows that the maturation delay plays an important role in the dynamical behavior of three species system. Also obtain some threshold values of maturation delay for the stability-switching of the particular system. In succession, using the normal form theory and center manifold argument, we derive the explicit formulas which determine the stability and direction of bifurcating periodic solutions. Finally, a numerical simulation for supporting the theoretical analysis is given.Comment: 28 pages, 9 figure

Evolutionary tradeoff and equilibrium in an aquatic predator-prey system

Due to the conventional distinction between ecological (rapid) and evolutionary (slow)timescales, ecological and population models to date have typically ignored the effects of evolution. Yet the potential for rapid evolutionary change has been recently established and may be critical to understanding how populations adapt to changing environments. In this paper we examine the relationship between ecological and evolutionary dynamics, focusing on a well-studied experimental aquatic predator-prey system (Fussmann et al. 2000; Shertzer et al. 2002; Yoshida et al. 2003). Major properties of predator-prey cycles in this system are determined by ongoing evolutionary dynamics in the prey population. Under some conditions, however, the populations tend to apparently stable steady-state densities. These are the subject of the present paper. We examine a previously developed model for the system, to determine how evolution shapes properties of the equilibria, in particular the number and identity of coexisting prey genotypes. We then apply these results to explore how evolutionary dynamics can shape the responses of the system to "management": externally imposed alterations in conditions. Specifically, we compare the behavior of the system including evolutionary dynamics, with predictions that would be made if the potential for rapid evolutionary change is negelected. Finally, we posit some simple experiments to verify our prediction that evolution can have significant qualitative effects on observed population-level responses to changing conditions.Comment: 30 pages including 8 figures, 2 tables and an Appendix; to appear in Bulletin of Mathematical Biology. Revised three Figures, added references and expanded Section

Bifurcation Analysis of Piecewise Smooth Ecological Models

The aim of this paper is the study of the long-term behavior of population communities described by piecewise smooth models (known as Filippov systems). Models of this kind are often used to describe populations with selective switching between alternative habitats or diets or to mimic the evolution of an exploited resource where harvesting is forbidden when the resource is below a prescribed threshold. The analysis is carried out by performing the bifurcation analysis of the model with respect to two parameters. A relatively simple method, called the puzzle method, is proposed to construct the complete bifurcation diagram step-by-step. The method is illustrated through four examples concerning the exploitation and protection of interacting populations

A prey-predator fishery model with endogenous switching of harvesting strategy

We propose a dynamic model to describe a fishery where both preys and predators are harvested by a population of fishermen who are allowed to catch only one of the two species at a time. According to the strategy currently employed by each agent, i.e. the harvested variety, at each time period the population of fishermen is partitioned into two groups, and an evolutionary mechanism regulates how agents dynamically switch from one strategy to the other in order to improve their profits. Among the various dynamic models proposed, the most realistic is a hybrid system formed by two ordinary differential equations, describing the dynamics of the interacting species under fishing pressure, and an impulsive variable that evolves in a discrete time scale, in order to describe the changes of the fraction of fishermen that harvest a given stock. The aim of the paper is to analyze the economic consequences of this kind of self-regulating fishery, as well as its biological sustainability, in comparison with other regulatory policies. Our analytic and numerical results give evidence that in some cases this kind of myopic, evolutionary self-regulation might ensure a satisfactory trade-off between profit maximization and resource conservation

Deterministic continutation of stochastic metastable equilibria via Lyapunov equations and ellipsoids

Numerical continuation methods for deterministic dynamical systems have been one of the most successful tools in applied dynamical systems theory. Continuation techniques have been employed in all branches of the natural sciences as well as in engineering to analyze ordinary, partial and delay differential equations. Here we show that the deterministic continuation algorithm for equilibrium points can be extended to track information about metastable equilibrium points of stochastic differential equations (SDEs). We stress that we do not develop a new technical tool but that we combine results and methods from probability theory, dynamical systems, numerical analysis, optimization and control theory into an algorithm that augments classical equilibrium continuation methods. In particular, we use ellipsoids defining regions of high concentration of sample paths. It is shown that these ellipsoids and the distances between them can be efficiently calculated using iterative methods that take advantage of the numerical continuation framework. We apply our method to a bistable neural competition model and a classical predator-prey system. Furthermore, we show how global assumptions on the flow can be incorporated - if they are available - by relating numerical continuation, Kramers' formula and Rayleigh iteration.Comment: 29 pages, 7 figures [Fig.7 reduced in quality due to arXiv size restrictions]; v2 - added Section 9 on Kramers' formula, additional computations, corrected typos, improved explanation