Modeling the effects of density dependent emigration, weak Allee effects, and matrix hostility on patch-level population persistence

James T. Cronin; Nalin Fonseka; Jerome Goddard II; Jackson Leonard; Ratnasingham Shivaji; James T. Cronin; Nalin Fonseka; Jerome Goddard II; Jackson Leonard; Ratnasingham Shivaji

doi:10.3934/mbe.2020090

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 2: 1718-1742. doi: 10.3934/mbe.2020090

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Research article

Modeling the effects of density dependent emigration, weak Allee effects, and matrix hostility on patch-level population persistence

1.
Department of Biological Sciences, Louisiana State University, Baton Rouge, LA 70803, USA
2.
Department of Mathematics and Statistics, University of North Carolina at Greensboro, Greensboro, NC 27412, USA
3.
Department of Mathematics and Computer science, University of Auburn Montgomery, Montgomery, AL 36117, USA

Received: 30 August 2019 Accepted: 17 November 2019 Published: 11 December 2019

The relationship between conspecific density and the probability of emigrating from a patch can play an essential role in determining the population-dynamic consequences of an Allee effect. In this paper, we model a population that inside a patch is diffusing and growing according to a weak Allee effect per-capita growth rate, but the emigration probability is dependent on conspecific density. The habitat patch is one-dimensional and is surrounded by a tuneable hostile matrix. We consider five different forms of density dependent emigration (DDE) that have been noted in previous empirical studies. Our models predict that at the patch-level, DDE forms that have a positive slope will counteract Allee effects, whereas, DDE forms with a negative slope will enhance them. Also, DDE can have profound effects on the dynamics of a population, including producing very complicated population dynamics with multiple steady states whose density profile can be either symmetric or asymmetric about the center of the patch. Our results are obtained mathematically through the method of subsuper solutions, time map analysis, and numerical computations using Wolfram Mathematica.
- habitat fragmentation,
- weak Allee effect,
- patch-level Allee effect,
- reaction diffusion model,
- density dependent emigration
Citation: James T. Cronin, Nalin Fonseka, Jerome Goddard II, Jackson Leonard, Ratnasingham Shivaji. Modeling the effects of density dependent emigration, weak Allee effects, and matrix hostility on patch-level population persistence[J]. Mathematical Biosciences and Engineering, 2020, 17(2): 1718-1742. doi: 10.3934/mbe.2020090

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Abstract

The relationship between conspecific density and the probability of emigrating from a patch can play an essential role in determining the population-dynamic consequences of an Allee effect. In this paper, we model a population that inside a patch is diffusing and growing according to a weak Allee effect per-capita growth rate, but the emigration probability is dependent on conspecific density. The habitat patch is one-dimensional and is surrounded by a tuneable hostile matrix. We consider five different forms of density dependent emigration (DDE) that have been noted in previous empirical studies. Our models predict that at the patch-level, DDE forms that have a positive slope will counteract Allee effects, whereas, DDE forms with a negative slope will enhance them. Also, DDE can have profound effects on the dynamics of a population, including producing very complicated population dynamics with multiple steady states whose density profile can be either symmetric or asymmetric about the center of the patch. Our results are obtained mathematically through the method of subsuper solutions, time map analysis, and numerical computations using Wolfram Mathematica.

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