Modeling metapopulation dynamics by using deterministic patch equilibrium and stochastic metapopulation extinction models

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Introduction
The success of conservation of a species depends on the understanding of the ecological factors driving the continued survival of that population (Bonsall et al. 2014). This will aid in the understanding of the temporal dynamics of the population when looking at the relationship between per capita growth rate and the local density of that population (Bonsall et al. 2014). When populations are fragmented, they rely on dispersal through migration to prevent inbreeding, and run the risk of extinction if they can’t (Bicknell et al. 2014). Populations are classified as metapopulations if they are demographically or genetically isolated from one another (Hanski 1998).
The aim of this study was to 1) determine how the proportion of habitat patches occupied influenced the change in patch incidence and 2) investigate how the extinction and migration rates influence the amount of patches occupied.
Methods and materials
Deterministic equilibrium model
A vensim model was set up according to the following equation where dP is the change in patch incidence, m is the rate of successful migration, P is the proportion of patches and E is the extinction rate of occupied patches: dP/ dt = mP (1-P)-EP

As m and E are the parameters of the population, they are entered as: m= 0,2 and E=0,1 and assumed to remain the same throughout the simulation. The proportion of patches is varied, first assuming that most patches are empty (P= 0,1) and then most patches are occupied (P=1,0). Graphs were then constructed for dP and P.
Stochastic metapopulation extinction model
For this model, it was assumed that the metapopulation consisted of 6 populations with identical extinction probabilities i...

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