A Predator–Prey Two-Sex Branching Process

Gutiérrez, Cristina; Minuesa, Carmen

doi:10.3390/math8091408

Cited by 2 publications

(3 citation statements)

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“…Before finishing this section, we establish the analogous of the extinction-explosion dichotomy, typical in branching process theory. The proof is omitted since it follows the same steps as the proof of Proposition 8 in [6].…”

Section: Biological Interpretationmentioning

confidence: 99%

“…Later, necessary and sufficient conditions for the fixation of both populations are studied for this model in [2]. Recently, a predator-prey two-sex branching process with promiscuous mating is introduced in [6] to model the interaction of predator and prey populations assuming that both species are formed by females and males having sexual reproduction. Necessary and sufficient conditions for the extinction of the population, the fixation of one of the species and the coexistence of both of them are also provided in this paper.…”

Section: Introductionmentioning

confidence: 99%

“…Another handicap of the models in [3] and [6] is that the prey population grows indefinitely in absence of predators. Although this is a behaviour observed in the simplest Lotka-Volterra model, an exponential growth does not seem to be a realistic feature because the limited environmental resources force preys to compete for available food.…”

Section: Introductionmentioning

confidence: 99%

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Predator–prey density-dependent branching processes

Gutiérrez

Minuesa

2022

Stochastic Models

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Two density-dependent branching processes are considered to model predatorprey populations. For both models, preys are considered to be the main food supply of predators. Moreover, in each generation the number of individuals of each species is distributed according to a binomial distribution with size given by the species population size and probability of success depending on the density of preys per predator at the current generation. The difference between the two proposed processes lies in the food supply of preys. In the first one, we consider that preys have all the food they need at their disposal while in the second one, we assume that the natural resources of the environment are limited and therefore there exists a competition among preys for food supplies. Results on the fixation and extinction of both species as well as conditions for the coexistence are provided for the first model. On the event of coexistence of both populations and on the prey fixation event, the limiting growth rates are obtained. For the second model, we prove that the extinction of the entire system occurs almost surely. Finally, the evolution of both models over the generations is illustrated by simulated examples. Those examples validate our analytical findings.

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Section: Biological Interpretationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%