262 research outputs found

    Effects of rapid prey evolution on predator-prey cycles

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    We study the qualitative properties of population cycles in a predator-prey system where genetic variability allows contemporary rapid evolution of the prey. Previous numerical studies have found that prey evolution in response to changing predation risk can have major quantitative and qualitative effects on predator-prey cycles, including: (i) large increases in cycle period, (ii) changes in phase relations (so that predator and prey are cycling exactly out of phase, rather than the classical quarter-period phase lag), and (iii) "cryptic" cycles in which total prey density remains nearly constant while predator density and prey traits cycle. Here we focus on a chemostat model motivated by our experimental system [Fussmann et al. 2000,Yoshida et al. 2003] with algae (prey) and rotifers (predators), in which the prey exhibit rapid evolution in their level of defense against predation. We show that the effects of rapid prey evolution are robust and general, and furthermore that they occur in a specific but biologically relevant region of parameter space: when traits that greatly reduce predation risk are relatively cheap (in terms of reductions in other fitness components), when there is coexistence between the two prey types and the predator, and when the interaction between predators and undefended prey alone would produce cycles. Because defense has been shown to be inexpensive, even cost-free, in a number of systems [Andersson and Levin 1999, Gagneux et al. 2006,Yoshida et al. 2004], our discoveries may well be reproduced in other model systems, and in nature. Finally, some of our key results are extended to a general model in which functional forms for the predation rate and prey birth rate are not specified.Comment: 35 pages, 8 figure

    A Universal Bifurcation Diagram for Seasonally Perturbed Predator-Prey Models

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    The bifurcations of a periodically forced predator-prey model (the chemostat model), with a prey feeding on a limiting nutrient, are numerically detected with a continuation technique. Eight bifurcation diagrams are produced (one for each parameter in the model) and shown to be topologically equivalent. These diagrams are also equivalent to those of the most commonly used predator-prey model (the Rosenzweig-McArthur model). Thus, all basic modes of behavior of the two main predator-prey models can be explained by means of a single bifurcation diagram

    Stability and Hopf Bifurcation Analysis of a Simple Nutrient- Prey-Predator Model with Intratrophic Predation in Chemostat

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    A 3-dimensional nutrient-prey-predator model with intratrophic predation is proposed and studied. Some elementary properties such as invariance of nonnegativity, boundedness and dissipativity of the system are presented. The purpose of this chapter is to study the existence and stability of equilibria along with the effects of intratrophic predation towards the positions and stability of those equilibria of the system. We also investigate the occurrence of Hopf bifurcation. In the case when there is no presence of predator organisms, intratrophic predation may not give impact on the stability of equilibria of the system. We also analysed global stability of the equilibrium point. A suitable Lyapunov function is defined for global stability analysis and some results of persistence analysis are presented for the existence of positive interior equilibrium point. Besides that, Hopf bifurcation analysis of the system are demonstrated

    Numerical equilibrium analysis for structured consumer resource models

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    In this paper, we present methods for a numerical equilibrium and stability analysis for models of a size structured population competing for an unstructured resource. We concentrate on cases where two model parameters are free, and thus existence boundaries for equilibria and stability boundaries can be defined in the (two-parameter) plane. We numerically trace these implicitly defined curves using alternatingly tangent prediction and Newton correction. Evaluation of the maps defining the curves involves integration over individual size and individual survival probability (and their derivatives) as functions of individual age. Such ingredients are often defined as solutions of ODE, i.e., in general only implicitly. In our case, the right-hand sides of these ODE feature discontinuities that are caused by an abrupt change of behavior at the size where juveniles are assumed to turn adult. So, we combine the numerical solution of these ODE with curve tracing methods. We have implemented the algorithms for “Daphnia consuming algae” models in C-code. The results obtained by way of this implementation are shown in the form of graphs

    Biological Models With Time Delay

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    This MQP can be divided into three main chapters. For each chapter, we studied three different mathematical models, which are the Logistic Growth model, the SIR model and the Chemostat model. These biological mathematical models all have time delay in part of their processes. Based on the specific phenomenological assumptions, we have different ways of building up the model through ordinary differential equations. The goal of this project is to investigate three models with time delay. We will also perform global sensitivity analysis for each model and study how uncertainty in the output of each model can be attributed to different sources of uncertainty in the model input. Applications for each model with time delay will be provided in this project with numerical results

    The 3-D bifurcation in a chemostat with n th and m th yields

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    The structure of solutions of the three dimensional chemostat competition system is analysed. The stability of equilibrium points and the three dimensional Hopf bifurcation of the system are discussed. The conditions of the existence of limit cycles on the two dimensional stable manifold when one microorganism vanishes are obtained. Some examples are used to show the applicability of the results

    Biological Models With Time Delay

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    This MQP can be divided into three main chapters. For each chapter, we studied three different mathematical models, which are the Logistic Growth model, the SIR model and the Chemostat model. These biological mathematical models all have time delay in part of their processes. Based on the specific phenomenological assumptions, we have different ways of building up the model through ordinary differential equations. The goal of this project is to investigate three models with time delay. We will also perform global sensitivity analysis for each model and study how uncertainty in the output of each model can be attributed to different sources of uncertainty in the model input. Applications for each model with time delay will be provided in this project with numerical results

    The 3-D bifurcation in a chemostat with n th and m th yields

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    The structure of solutions of the three dimensional chemostat competition system is analysed. The stability of equilibrium points and the three dimensional Hopf bifurcation of the system are discussed. The conditions of the existence of limit cycles on the two dimensional stable manifold when one microorganism vanishes are obtained. Some examples are used to show the applicability of the results

    A predator-prey model in the chemostat with time delay

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    Multiple Attractors, Catastrophes and Chaos in Seasonally Perturbed Predator-Prey Communities

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    The classical predator-prey model is considered in this paper with reference to the case of periodically varying parameters. Six elementary seasonality mechanisms are identified and analyzed in detail by means of a continuation technique producing complete bifurcation diagrams. The results show that each elementary mechanism can give rise to multiple attractors and that catastrophic transitions can occur when suitable parameters are slightly changed. Moreover, the two classical routes to chaos, namely, torus destruction and cascade of period doublings, are numerically detected. Since in the case of constant parameters the model cannot have multiple attractors, catastrophes, and chaos, the results support the conjecture that seasons can very easily give rise to complex population dynamics
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