On the Properties of a Class of Impulsive Competition Beverton–Holt Equations

Manuel De La Sen,Aitor J Garrido,Asier Ibeas,Izaskun Garrido,Santiago Alonso-Quesada

doi:10.3390/app11199020

Manuel De La Sen, Aitor J Garrido + Show 3 more

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https://doi.org/10.3390/app11199020

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Abstract

This paper is devoted to a type of combined impulsive discrete Beverton–Holt equations in ecology when eventual discontinuities at sampling time instants are considered. Such discontinuities could be interpreted as impulses in the corresponding continuous-time logistic equations. The set of equations involve competition-type coupled dynamics among a finite set of species. It is assumed that, in general, the intrinsic growth rates and the carrying capacities are eventually distinct for the various species. The impulsive parts of the equations are parameterized by harvesting quotas and independent consumptions which are also eventually distinct for the various species and which control the populations’ evolution. The performed study includes the existence of extinction and non-extinction equilibrium points, the conditions of non-negativity and boundedness of the solutions for given finite non-negative initial conditions and the conditions of asymptotic stability without or with extinction of the solutions.

Highlights

The above artificial sequence allows an easy integration of the eventual independent consumption contribution to the stability of non-extinction equilibrium points
This paper discusses extinction and non-extinction conditions obtained from an imimpulsive-type competition Beverton–Holt equation, which, in the most general case, pulsive-type competition Beverton–Holt equation, which, in the most general case, is is modeled under a time-varying parameterization
Constraints the harvesting, tion/non-extinction conditions are testable constraints on the harvesting, typically ically being hunting/fishing quotas, or, alternatively, on the independent consumption, being hunting/fishing quotas, or, alternatively, on the independent consumption, which which are the relevant parameters in the discontinuities at sampling time instants of the are the relevant parameters in the discontinuities at sampling time instants of the populapopulation dynamics

Summary

Introduction

Its typical usefulness is related to the evolution of some species which reproduce by eggs. Some usefulness of the models has been pointed out related to the fishery industry relying on the exploitation of species such as, for instance, coho salmon, plaice, haddock and others [1]. The discretization could potentially be extended to time intervals, including two samples of, in general, distinct sampling periods, to separate the evolution of egg/larvae from that of reproductive adults in some species. Two different sampling periods can be involved when the stages of egg/larvae evolution and those of adult evolution have different relevant average time intervals. An important effort has been devoted to rigorously prove the so-called Cushing–Henson

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