Extinction probabilities in predator–prey models

Two stochastic models are developed for the predator-prey process. In each case it is shown that ultimate extinction of one of the two populations is certain to occur in finite time. For each model an exact expression is derived for the probability that the predators eventually become extinct when the prey birth rate is 0. These probabilities are used to derive power series approximations to extinction probabilities when the prey birth rate is not 0. On comparison with values obtained by numerical analysis the approximations are shown to be very satisfactory when initial population sizes and prey birth rate are all small. An approximation to the mean number of changes before extinction occurs is also obtained for one of the models.

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Extinction times for certain predator–prey processes

Journal of Applied Probability ◽

10.2307/3213988 ◽

1988 ◽

Vol 25 (3) ◽

pp. 612-616 ◽

Cited By ~ 3

Author(s):

C. J. Ridler-Rowe

Keyword(s):

Initial Population ◽

Predator Prey ◽

The Mean ◽

Predator Prey Models

Finiteness of mean extinction times for certain predator-prey models has been established by Hitchcock (1986) with the aid of a criterion of Reuter (1957). Using this criterion and a ‘minimisation' lemma this note shows that the mean extinction times tend to zero as the combined initial population of predators and prey becomes large.

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Extinction times for certain predator–prey processes

Journal of Applied Probability ◽

10.1017/s0021900200041310 ◽

1988 ◽

Vol 25 (03) ◽

pp. 612-616

Author(s):

C. J. Ridler-Rowe

Keyword(s):

Initial Population ◽

Predator Prey ◽

The Mean ◽

Predator Prey Models

Finiteness of mean extinction times for certain predator-prey models has been established by Hitchcock (1986) with the aid of a criterion of Reuter (1957). Using this criterion and a ‘minimisation' lemma this note shows that the mean extinction times tend to zero as the combined initial population of predators and prey becomes large.

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The global dynamics of stochastic Holling type II predator-prey models with non constant mortality rate

Filomat ◽

10.2298/fil1718811z ◽

2017 ◽

Vol 31 (18) ◽

pp. 5811-5825

Author(s):

Xinhong Zhang

Keyword(s):

Mortality Rate ◽

Stochastic Model ◽

Sufficient Conditions ◽

Positive Periodic Solution ◽

Global Dynamics ◽

Type Ii ◽

Predator Prey ◽

Persistence In The Mean ◽

The Mean ◽

Predator Prey Models

In this paper we study the global dynamics of stochastic predator-prey models with non constant mortality rate and Holling type II response. Concretely, we establish sufficient conditions for the extinction and persistence in the mean of autonomous stochastic model and obtain a critical value between them. Then by constructing appropriate Lyapunov functions, we prove that there is a nontrivial positive periodic solution to the non-autonomous stochastic model. Finally, numerical examples are introduced to illustrate the results developed.

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Coexistence and Noncoexistence of Markovian Viruses and their Hosts

Journal of Applied Probability ◽

10.1239/jap/1395771423 ◽

2014 ◽

Vol 51 (1) ◽

pp. 191-208 ◽

Cited By ~ 2

Author(s):

Jakob E. Björnberg ◽

Erik I. Broman

Keyword(s):

Virus Infection ◽

Predator Prey ◽

Classic Problem ◽

Infected Cells ◽

Predator Prey Models ◽

Two Populations ◽

Competing Populations

Examining possibilities for the coexistence of two competing populations is a classic problem which dates back to the earliest ‘predator-prey’ models. In this paper we study this problem in the context of a model introduced in Björnberg et al. (2012) for the spread of a virus infection in a population of healthy cells. The infected cells may be seen as a population of ‘predators’ and the healthy cells as a population of ‘prey’. We show that, depending on the parameters defining the model, there may or may not be coexistence of the two populations, and we give precise criteria for this.

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Coexistence and Noncoexistence of Markovian Viruses and their Hosts

Journal of Applied Probability ◽

10.1017/s0021900200010172 ◽

2014 ◽

Vol 51 (01) ◽

pp. 191-208 ◽

Cited By ~ 1

Author(s):

Jakob E. Björnberg ◽

Erik I. Broman

Keyword(s):

Virus Infection ◽

Predator Prey ◽

Classic Problem ◽

Infected Cells ◽

Predator Prey Models ◽

Two Populations ◽

Competing Populations

Examining possibilities for the coexistence of two competing populations is a classic problem which dates back to the earliest ‘predator-prey’ models. In this paper we study this problem in the context of a model introduced in Björnberg et al. (2012) for the spread of a virus infection in a population of healthy cells. The infected cells may be seen as a population of ‘predators’ and the healthy cells as a population of ‘prey’. We show that, depending on the parameters defining the model, there may or may not be coexistence of the two populations, and we give precise criteria for this.

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Birth Rate Effects on an Age-Structured Predator-Prey Model with Cannibalism in the Prey

Abstract and Applied Analysis ◽

10.1155/2015/241312 ◽

2015 ◽

Vol 2015 ◽

pp. 1-7 ◽

Cited By ~ 2

Author(s):

Francisco J. Solis ◽

Roberto A. Ku-Carrillo

Keyword(s):

Birth Rate ◽

Birth Rates ◽

Differential Systems ◽

Predator Prey ◽

Predator Prey Model ◽

New Birth ◽

Rate Effects ◽

Stable Equilibria ◽

Predator Prey Models ◽

Age Structured

We develop a family of predator-prey models with age structure and cannibalism in the prey population. It consists of systems ofmordinary differential equations, wheremis a parameter associated with new proposed prey birth rates. We discuss how these new birth rates give the required flexibility to produce differential systems with well-behaved solutions. The main feature required in these models is the coexistence among the involved species, which translates mathematically into stable equilibria and periodic solutions. The search for such characteristics is based on heuristic predation functions that account for cannibalism in the prey.

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COMPLEX BEHAVIOR IN SIMPLE MODELS OF BIOLOGICAL COEVOLUTION

International Journal of Modern Physics C ◽

10.1142/s012918310901445x ◽

2009 ◽

Vol 20 (09) ◽

pp. 1387-1397 ◽

Cited By ~ 1

Author(s):

PER ARNE RIKVOLD

Keyword(s):

Kinetic Monte Carlo ◽

Dynamical Behavior ◽

Individual Species ◽

Predator Prey ◽

Population Sizes ◽

Kinetic Monte Carlo Simulations ◽

Predator Prey Models ◽

Power Law Distributions ◽

Adaptive Foraging ◽

Type Ii Functional Response

We explore the complex dynamical behavior of simple predator-prey models of biological coevolution that account for interspecific and intraspecific competition for resources, as well as adaptive foraging behavior. In long kinetic Monte Carlo simulations of these models we find quite robust 1/f-like noise in species diversity and population sizes, as well as power-law distributions for the lifetimes of individual species and the durations of quiet periods of relative evolutionary stasis. In one model, based on the Holling Type II functional response, adaptive foraging produces a metastable low-diversity phase and a stable high-diversity phase.

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