Ecoepidemic models with prey group defense and feeding saturation

Abstract In this paper we consider a model for the herd behavior of prey, that are subject to attacks by specialist predators. The latter are affected by a transmissible disease. With respect to other recently introduced models of the same nature, we focus here our attention to the possible feeding satiation phenomenon. The system dynamics is thoroughly investigated, to show the occurrence of several types of bifurcations. In addition to the transcritical and Hopf bifurcation that occur commonly in predator–prey system also a zero-Hopf and a global bifurcation occur. The Hopf and the global bifurcation occur only in the disease-free (so purely demographic) system. The latter is a heteroclinic connection for the between saddle equilibrium points where a stable limit cycle is disrupted and where the system disease-free collapses while in a parameter space region the endemic system exists stably.

[1]  E. Venturino,et al.  A comparison of several plankton models for red tides , 2014 .

[2]  Marina Schroder,et al.  Nonlinear Dynamics Of Interacting Populations , 2016 .

[3]  Joydev Chattopadhyay,et al.  A predator—prey model with disease in the prey , 1999 .

[4]  G. Quispel,et al.  The saddle-node-transcritical bifurcation in a population model with constant rate harvesting , 2010, 1002.4150.

[5]  Sergei Petrovskii,et al.  Spatiotemporal behavior of a prey–predator system with a group defense for prey , 2013 .

[6]  Ezio Venturino,et al.  The Influence of Diseases on Lotka-Volterra Systems , 1993 .

[7]  Stephen Wiggins.  Global Bifurcations and Chaos: Analytical Methods , 1988 .

[8]  H. I. Freedman,et al.  Predator-prey populations with parasitic infection , 1989, Journal of mathematical biology.

[9]  M. Boer,et al.  Ecological consequences of global bifurcations in some food chain models. , 2010, Mathematical biosciences.

[10]  G S Wolkowicz,et al.  Predator-prey systems with group defence: The paradox of enrichment revisited , 1986, Bulletin of mathematical biology.

[11]  Sergei Petrovskii,et al.  Spatiotemporal patterns in ecology and epidemiology , 2007 .

[12]  Herbert W. Hethcote,et al.  Dynamic models of infectious diseases as regulators of population sizes , 1992, Journal of mathematical biology.

[13]  C. S. Holling.  The functional response of invertebrate predators to prey density , 1966 .

[14]  Ezio Venturino,et al.  Patchy agglomeration as a transition from monospecies to recurrent plankton blooms. , 2008, Journal of theoretical biology.

[15]  S. Busenberg,et al.  Analysis of a disease transmission model in a population with varying size , 1990, Journal of mathematical biology.

[16]  Willy Govaerts,et al.  MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs , 2003, TOMS.

[17]  D. DeAngelis,et al.  Effects of spatial grouping on the functional response of predators. , 1999, Theoretical population biology.

[18]  James T. Tanner,et al.  THE STABILITY AND THE INTRINSIC GROWTH RATES OF PREY AND PREDATOR POPULATIONS , 1975 .

[19]  George A K van Voorn,et al.  Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong Allee effect. , 2007, Mathematical biosciences.

[20]  Ezio Venturino,et al.  A MINIMAL MODEL FOR ECOEPIDEMICS WITH GROUP DEFENSE , 2011 .

[21]  Ezio Venturino,et al.  Modeling herd behavior in population systems , 2011 .

[22]  S. Wiggins.  Introduction to Applied Nonlinear Dynamical Systems and Chaos , 1989 .

[23]  B. Kooi,et al.  Smoking epidemic eradication in a eco-epidemiological dynamical model , 2013 .

[24]  Ezio Venturino,et al.  An ecoepidemic model with diseased predators and prey group defense , 2013, Simul. Model. Pract. Theory.

[25]  Herbert W. Hethcote,et al.  The Mathematics of Infectious Diseases , 2000, SIAM Rev..

[26]  B W Kooi,et al.  Numerical Bifurcation Analysis of Ecosystems in a Spatially Homogeneous Environment , 2003, Acta biotheoretica.

[27]  Ezio Venturino,et al.  The role of transmissible diseases in the Holling-Tanner predator-prey model. , 2006, Theoretical population biology.

[28]  Peter A. Braza,et al.  Predator–prey dynamics with square root functional responses , 2012 .

[29]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[30]  T. O. Carroll,et al.  Modeling the role of viral disease in recurrent phytoplankton blooms , 1994 .

[31]  Y. Kuznetsov.  Elements of Applied Bifurcation Theory , 2023, Applied Mathematical Sciences.