Bifurcation Analysis of a Predator-Prey System with Nonmonotonic Functional Response

Abstract

Abstract. We consider a predator-prey system with nonmonotonic functional response: p(x) = mx ax 2 +bx+1. By allowing b to be negative (b > −2 √ a), p(x) is concave up for small values of x > 0 as it is for the sigmoidal functional response. We show that in this case there exists a Bogdanov-Takens bifurcation point of codimension 3, which acts as an organizing center for the system. We study the Hopf and homoclinic bifurcations and saddle-node bifurcation of limit cycles of the system. We also describe the bifurcation sequences in each subregion of parameter space as the death rate of the predator is varied. In contrast with the case b ≥ 0, we prove that when −2 √ a < b < 0, a limit cycle can coexist with a homoclinic loop. The bifurcation sequences involving Hopf bifurcations, homoclinic bifurcations, as well as the saddle-node bifurcations of limit cycles are determined using information from the complete study of the Bogdanov-Takens bifurcation point of codimension 3 and the geometry of the system. Examples of the predicted bifurcation curves are also generated numerically using XPPAUT. Our work extends the results in [F. Rothe and D. S. Shafer, Proc. Roy.

Nonlinear Dynamics of a Controlled Stirred Tank Bioreactor With Predator-Prey Relationship

“…Then predator-prey biocenosis develops in a bioreactor. This phenomenon has a significant impact on the stationary and dynamic characteristics of a bioreactor (Alhumazi and Ajbar., 2005;Butler et al, 1983;El-Owaidy and Moniem, 2003;Li and Kuang, 2000; 350 cpe.czasopisma.pan.pl; degruyter.com/view/j/cpe Moghadas and Gumel, 2003;Zhu et al, 2002). The characteristic sustained oscillations have been also observed experimentally (Tsuchiya et al, 1972).…”

On the Dynamical Behavior of a Two-Prey One-Predator System with Two-Type Functional Responses

“…In fact, there are many researchers who studied this subsystem about dynamical behaviors(cf, [12,18,26] ). Especially, Ruan and Xiao in [18] investigated the equilibrium points and their stability.…”

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“…[Баутин, Леонтович, 1990;Bazykin, 1998;Broer, Naudot, Roussarie, Saleh, 2007;Broer, Gaiko, 2010;Gaiko, 2003;Гайко, 2011;Lamontagne, Coutu, Rousseau, 2008;Zhu, Campbell, Wolkowicz, 2002].…”

“…Однако было бы более реалистично рассматривать нелинейные и ограниченные функции отклика, и в литературе действительно используются различные виды таких функций для мо-делирования отклика хищников; см. [Bazykin, 1998;Broer, Naudot, Roussarie, Saleh, 2007;Broer, Gaiko, 2010;Гайко, 2011;Holling, 1959;Lamontagne, Coutu, Rousseau, 2008;Zhu, Campbell, Wolkowicz, 2002].…”

“…В отличие от случая обобщенной функции Холлинга IV типа, где функция отклика стремится к нулю при стремлении популяции жертв к бесконечности (см. [Broer, Naudot, Roussarie, Saleh, 2007;Broer, Gaiko, 2010;Гайко, 2011;Zhu, Campbell, Wolkowicz, 2002]), обобщенная функция Холлинга III типа стремится к ненулевому значению при стремлении популяции жертв к бесконечности. При этом функция отклика III типа имеет максимум в некоторой точке при 0 < b [ Lamontagne, Coutu, Rousseau, 2008].…”