Ratio-Dependent Predator-Prey Models of Interacting Populations

Ratio-dependent predator-prey models are increasingly favored by both the theoretical and experimental ecologists as a more suitable alternative to describe predator-prey interactions when the predators hunt seriously. In this article, the classical Bazykin’s model is modified with ratio-dependent functional response. Stability and bifurcation situations of the system are observed. Since the ratio-dependent model always has difficult dynamics in the vicinity of the origin, the analytical behavior of the system near origin is studied completely. It is found that paradox of enrichment can happen to this system under certain parameter values, although the functional response is ratio-dependent. The parametric space for Turing spatial structure is determined. We also conclude that competition among the predator population might be beneficial for predator species under certain circumstances. Finally, ecological interpretations of our results are presented in the discussion section.

[1]  Ezio Venturino,et al.  An ecoepidemiological model with disease in predator: the ratio‐dependent case , 2007 .

[2]  R. May,et al.  Stability and Complexity in Model Ecosystems , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[3]  Sze-Bi Hsu,et al.  Rich dynamics of a ratio-dependent one-prey two-predators model , 2001, Journal of mathematical biology.

[4]  A. Gutierrez.  Physiological Basis of Ratio-Dependent Predator-Prey Theory: The Metabolic Pool Model as a Paradigm , 1992 .

[5]  Roger Arditi,et al.  Ratio-Dependent Predation: An Abstraction That Works , 1995 .

[6]  C Jost,et al.  Identifying predator-prey processes from time-series. , 2000, Theoretical population biology.

[7]  Sze-Bi Hsu,et al.  A ratio-dependent food chain model and its applications to biological control. , 2003, Mathematical biosciences.

[8]  Frederic Bartumeus,et al.  MUTUAL INTERFERENCE BETWEEN PREDATORS CAN GIVE RISE TO TURING SPATIAL PATTERNS , 2002 .

[9]  Peter A. Abrams,et al.  The Fallacies of "Ratio‐Dependent" Predation , 1994 .

[10]  Alekseev Vv.  [Effect of the saturation factor on population dynamics in the system prey-predator]. , 1973 .

[11]  Wayne M. Getz,et al.  Population Dynamics: a per capita Resource Approach , 1984 .

[12]  Yang Kuang,et al.  Global qualitative analysis of a ratio-dependent predator–prey system , 1998 .

[13]  David Greenhalgh,et al.  A predator–prey model with disease in the prey species only , 2007 .

[14]  Christian Jost,et al.  About deterministic extinction in ratio-dependent predator-prey models , 1999 .

[15]  Bernd Krauskopf,et al.  Nonlinear Dynamics of Interacting Populations , 1998 .

[16]  Joydev Chattopadhyay,et al.  Infection in prey population may act as a biological control in ratio-dependent predator?prey models , 2004 .

[17]  R. Arditi,et al.  From pattern to process: identifying predator–prey models from time-series data , 2001, Population Ecology.

[18]  I. Hanski.  The functional response of predators: Worries about scale , 1991 .

[19]  L. Luckinbill,et al.  Coexistence in Laboratory Populations of Paramecium Aurelia and Its Predator Didinium Nasutum , 1973 .

[20]  S. Hsu,et al.  Global analysis of the Michaelis–Menten-type ratio-dependent predator-prey system , 2001, Journal of mathematical biology.

[21]  Lansun Chen,et al.  A ratio-dependent predator-prey model with disease in the prey , 2002, Appl. Math. Comput..

[22]  Dongmei Xiao,et al.  Global dynamics of a ratio-dependent predator-prey system , 2001, Journal of mathematical biology.

[23]  R. Arditi,et al.  Coupling in predator-prey dynamics: Ratio-Dependence , 1989 .

[24]  R Arditi,et al.  Parametric analysis of the ratio-dependent predator–prey model , 2001, Journal of mathematical biology.

[25]  J. Hale,et al.  Ordinary Differential Equations , 2019, Fundamentals of Numerical Mathematics for Physicists and Engineers.

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

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