Journal of innovative applied mathematics and computational sciences
Volume 2, Numéro 3, Pages 41-68
2022-12-31
Effect Of Dispersal In Two-patch Environment With Richards Growth On Population Dynamics
Authors : Elbetch Bilel .
Abstract
In this paper, we consider a two-patch model coupled by migration terms,where each patch follows a Richards law. First, we prove the global stability of themodel. Second, in the case when the migration rate tends to infinity, the total carryingcapacity is given, which in general is different from the sum of the two carrying ca-pacities and depends on the parameters of the growth rate and also on the migrationterms. Using the theory of singular perturbations, we give an approximation of thesolutions of the system in this case. Finally, we determine the conditions under whichfragmentation and migration can lead to a total equilibrium population which mightbe greater or smaller than the sum of two carrying capacities and we give a completeclassification for all possible cases. The total equilibrium population formula for alarge migration rate plays an important role in this classification. We show that thischoice of local dynamics has an influence on the effect of dispersal. Comparing thedynamics of the total equilibrium population as a function of the migration rate withthat of the logistic model, we obtain the same behavior. In particular, we have only three situations that the total equilibrium population can occur: it is always greaterthan the sum of two carrying capacities, always smaller, and a third case, where theeffect of dispersal is beneficial for lower values of the migration rate and detrimentalfor the higher values. We end by examining the two-patch model where one growth rate is much larger than the second one, we compare the total equilibrium populationwith the sum of the two carrying capacities.
Keywords
Population dynamics, Richards Model, Asymmetric dispersal, SingularPerturbation
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