On the Gause predator–prey model with a refuge: A fresh look at the history

Abstract

Cite this article as: Vlastimil Křivan, On the gause predator-prey model with a refuge: a fresh look at the history, Journal of Theoretical Biology, doi:10.1016Biology, doi:10. /j.jtbi.2011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.On the Gause predator-prey model with a refuge: A fresh look at the history AbstractThis article re-analyses a prey-predator model with a refuge introduced by one of the founders of population ecology G. F. Gause and his co-workers to explain discrepancies between their observations and predictions of the Lotka-Volterra prey-predator model. They replaced the linear functional response used by Lotka and Volterra by a saturating functional response with a discontinuity at a critical prey density. At concentrations below this critical density prey were effectively in a refuge while at a (1) trajectories converge to a limit cycle, as predicted by F. G. Gause, (2) trajectories converge to an equilibrium, or (3) the prey population escapes predator control and grows to infinity.

A Thermostatted Kinetic Framework with Particle Refuge for the Modeling of Tumors Hiding

“…This section is meant to analytical results on the mathematical framework (13). In particular we define the density of the non-refusing particles of the functional subsystem f 1 as follows:…”

“…The thermostatted framework with particles refuge (13) acts as paradigms for the derivation of models. The derivation of a specific model means providing, by a suitable phenomenological interpretation of the system, a mathematical description of the microscopic interactions among the selected functional subsystems.…”

“…In the context of predator-prey frameworks, the dynamics and bifurcations analysis of mathematical models with refuge have gained much attention, see, among others, papers [10,11,12,13,14,15,16] and the review paper [17]. The use of refuge can be broadly defined to include any strategy that decreases predation risk, e.g., spatial or temporal refuges, prey aggregations, or reduced search activity by prey.…”

Sliding Bifurcation Analysis and Global Dynamics for a Filippov Predator-prey System

“…Recently the classical Lotka-Volterra model was extended by using a piecewise saturating function to replace the linear consumption rate for considering the observed experimental results theoretically [17,24]. For simplicity, denote H(Z) = x − ET with Z = (x, y) T ∈ R 2 + , where ET describes the critical prey population threshold, and the parameter ε can be defined as follows ε = 0, H(Z) = x − ET < 0, 1, H(Z) = x − ET > 0.…”

“…Therefore, if the density of the prey population falls below the threshold ET , that is, H(Z) < 0, then ε = 0, which indicates that the prey may avoid the predator via a habitat shift by moving to the refuge and the density of the predator will decrease [2]; if the density of the prey population increases and exceeds the threshold ET , that is, H(Z) > 0, then ε = 1, which indicates that the prey population may re-appear and once again become accessible to predators [2,16,17,24]. According to the above definition, the extended classical Lotka-Volterra model with a piecewise saturating function can be defined as the following Filippov system        dx(t) dt = rx(t) − εbx(t)y(t) 1 + bhx(t) , dy(t) dt = εkbx(t)y(t) 1 + bhx(t) − δy(t),…”