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Modeling effects of matrix heterogeneity on population persistence at the patch-level
Modeling effects of matrix heterogeneity on population persistence at the patch-level
N. Fonseka, A. Henderson, J. Goddard II, D. Nichols, R. Shivaji. MBE. May 2022
Habitat loss and fragmentation is the largest contributing factor to species extinction and declining biodiversity. Landscapes are becoming highly spatially heterogeneous with varying degrees of human modification. Much theoretical study of habitat fragmentation has historically focused on a simple theoretical landscape with patches of habitat surrounded by a spatially homogeneous hostile matrix. However, terrestrial habitat patches are often surrounded by complex mosaics of many different land cover types, which are rarely ecologically neutral or completely inhospitable environments. We employ an extension of a reaction diffusion model to explore effects of heterogeneity in the matrix immediately surrounding a patch in a one-dimensional theoretical landscape. Exact dynamics of a population exhibiting logistic growth, an unbiased random walk in the patch and matrix, habitat preference at the patch/matrix interface, and two functionally different matrix types for the one-dimensional landscape is obtained. These results show existence of a minimum patch size (MPS), below which population persistence is not possible. This MPS can be estimated via empirically derived estimates of patch intrinsic growth rate and diffusion rate, habitat preference, and matrix death and diffusion rates. We conclude that local matrix heterogeneity can greatly change model predictions, and argue that conservation strategies should not only consider patch size, configuration, and quality, but also quality and spatial structure of the surrounding matrix.
On the effects of density-dependent emigration on ecological models with logistic and weak Allee type growth terms
On the effects of density-dependent emigration on ecological models with logistic and weak Allee type growth terms
Acharya, A., Fonseka, N., Goddard II, J., Henderson, A., Shivaji, R.
Logistic-type growth (LTG) assumes a strictly negative density dependence between density and per-capita growth rate. Notwithstanding, Allee effects, the positive effects of increasing density on fitness, have been observed empirically in the literature since they were first described in the early 1930's for cooperatively breeding species. Though difficult to detect, empirical support for Allee effects spans a wide diversity of taxa. A common cause of an Allee effect in fitness is thought to be due to scarcity of reproductive opportunity at low densities. In the context of landscape ecology, Allee effects are particularly important and there is a growing list of studies that have examined the interplay between Allee effects and dispersal, broadly defined as movement between habitat patches. An Allee effect is considered strong if per-capita growth rate is negative for small densities, and weak otherwise (in this case, WAG). It is well known that population models with strong Allee effect growth will predict existence of a density threshold for which the population must remain above in order for persistence to be ensured. However, a weak Allee effect growth is not sufficient to ensure existence of such a threshold.
quad_4(5).pdfquad_5(5).pdfSigma-shaped bifurcation curves influenced by nonlinear boundary conditions for classes of reaction diffusion systems
Sigma-shaped bifurcation curves influenced by nonlinear boundary conditions for classes of reaction diffusion systems
Acharya, A., Fonseka, N., Henderson, A., Shivaji, R.
We analyze positive solutions to steady state reaction diffusion systems of the form:
A Sigma-shaped bifurcation curve for a class of reaction diffusion equations and an application to an ecological model
A Sigma-shaped bifurcation curve for a class of reaction diffusion equations and an application to an ecological model
Acharya, A.,Cronin, J.T., Fonseka, N., Goddard II, J., Henderson, A., Munoz, V., Shivaji, R.
We analyze the structure of positive solutions to a class of steady state equations arising in modeling a prey population that grows logistically, experiences Holling Type III predation from a generalist predator, and exhibits an overall negative relationship between density and emigration rate (-DDE). In particular, the predator is assumed to operate on a different time scale than the prey and, thus, its density is held constant. Under some general hypotheses on the reaction term and boundary nonlinearity, we establish existence, nonexistence, and multiplicity results for certain ranges of a parameter which is proportional to patch size squared via the method of sub-super-solutions. In particular, we establish that the bifurcation curve of positive solutions for the steady state equation is at least $\Sigma$-shaped. In this case, there is a range of patch size where a patch-level Allee effect occurs, i.e., a situation where the trivial solution and at least one other positive steady state are stable arises for small patch sizes, and a non-Allee effect type bi-stability arises for a range of larger patch sizes. As an application of our result, we consider the case when $\Omega$ is a ball, the reaction term is exactly logistic growth with a Type III functional response and the boundary nonlinearity is a -DDE form with a fast decay rate and show that the hypotheses in our theorems are satisfied. Further, when $\Omega = (0,1),$ we employ quadrature methods and computations using Wolfram Mathematica to show that the bifurcation diagram for positive solutions of this example is exactly $\Sigma$-shaped for certain values of the parameters. The occurrence of multiple steady states in real-world metapopulations can influence the fraction and distribution of occupied patches and cause uncertainty in predicting minimum patch size and density-area relationships.
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