Cancer immunoediting: A process driven by metabolic competition as a predator–prey–shared resource type model

In this paper, a Kolmogorov-type model, which includes the Gause-type model (Kuang and Freedman, 1988), the general predator-prey model (Huang 1988, Huang and Merrill 1989), and many other specialized models, is studied. The stability of equilibrium points, the existence and uniqueness of limit cycles in the model are proved.

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Analysis of a Nonautonomous Delayed Predator-Prey System with a Stage Structure for the Predator in a Polluted Environment

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2010/891812 ◽

2010 ◽

Vol 2010 ◽

pp. 1-18 ◽

Cited By ~ 2

Author(s):

G. P. Samanta

Keyword(s):

Sufficient Conditions ◽

Stage Structure ◽

Positive Periodic Solution ◽

Type Model ◽

Predator Population ◽

Constant Delay ◽

Volterra Type ◽

Globally Asymptotically Stable ◽

Predator Prey ◽

Prey System

A two-species nonautonomous Lotka-Volterra type model with diffusional migration among the immature predator population, constant delay among the matured predators, and toxicant effect on the immature predators in a nonprotective patch is proposed. The scale of the protective zone among the immature predator population can be regulated through diffusive coefficientsDi(t),i=1,2. It is proved that this system is uniformly persistent (permanence) under appropriate conditions. Sufficient conditions are derived to confirm that if this system admits a positive periodic solution, then it is globally asymptotically stable.

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Qualitative effects of introducing nonlinear birth and death rates for the predator in a predator-prey type model

BIOMATH ◽

10.11145/j.biomath.2017.03.167 ◽

2017 ◽

Vol 6 (1) ◽

pp. 1703167

Author(s):

Tihomir B. Ivanov ◽

Neli S. Dimitrova

Keyword(s):

Equilibrium Points ◽

Classical Model ◽

Uniform Boundedness ◽

Prey Type ◽

Type Model ◽

Qualitative Behavior ◽

Death Rates ◽

Predator Prey ◽

Stability And Bifurcations ◽

Predator Behavior

In this paper, we study how introducing nonlinear birth and death rates for the predator might affect the qualitative behavior of a mathematical model, describing predator-prey systems. We base our investigations on a known model, exhibiting anti-predator behavior. We propose a generalization of the latter by introducing generic birth and death rates for the predator and study the dynamics of the resulting system. We establish existence and uniqueness of positive model solutions, their uniform boundedness, existence, local stability and bifurcations of equilibrium points as well as global stability properties of the solutions. Most of the solution properties are demonstrated numerically and graphically by various numerical examples. Based on the obtained results, we show that the model with nonlinear birth and death rates can describe a much more complex behavior of the predator-prey system than the classical model (i.e., with linear rates) does.

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Deterministic models and identification of their parameters

Tatra Mountains Mathematical Publications ◽

10.2478/v10127-011-0032-2 ◽

2011 ◽

Vol 50 (1) ◽

pp. 1-11

Author(s):

Václav Pink

Keyword(s):

Simulated Data ◽

Point Of View ◽

Type Model ◽

Deterministic Models ◽

Data Set ◽

Predator Prey ◽

R Language ◽

Interior Equilibrium ◽

Coefficients Of Variation ◽

Prey Interaction

ABSTRACT This article deals with a possibility to identify parameters of a selected growth model of two populations coupled by a predator-prey interaction from a set of observed data. It starts with a brief description of the Gause-type model and of a property (interior equilibrium stability) important from a point of view of an application. Subsequently, data for four forms of the trophic function are simulated and then, a noise was added to the simulated data such that the coefficients of variation equal to 0.2, 0.3 and 0.4. For each data set, the parameters are estimated using a procedure implemented in the R-language package and the coordinates of equilibrium are computed. Then we can evaluate the effect of changing variation to the values of parameters and (un)stability of the equilibrium.

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