scholarly journals A mathematical study of diffusive logistic equations with mixed type boundary conditions

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Kazuaki Taira
Keyword(s):  
Mixed Type ◽  
Diffusion Rate ◽  
Critical Value ◽  
Initial Population ◽  
First Eigenvalue ◽  
Logistic Equations ◽  
Indefinite Weights ◽  
Recent Developments ◽  
Biological Interpretation ◽  
Type Boundary

<p style='text-indent:20px;'>The purpose of this paper is to provide a careful and accessible exposition of static bifurcation theory for a class of mixed type boundary value problems for diffusive logistic equations with indefinite weights, which model population dynamics in environments with spatial heterogeneity. We discuss the changes that occur in the structure of the positive solutions as a parameter varies near the first eigenvalue of the linearized problem, and prove that the most favorable situations will occur if there is a relatively large favorable region (with good resources and without crowding effects) located some distance away from the boundary of the environment. A biological interpretation of main theorem is that an initial population will grow exponentially until limited by lack of available resources if the diffusion rate is below some critical value; this idea is generally credited to the English economist T. R. Malthus. On the other hand, if the diffusion rate is above this critical value, then the model obeys the logistic equation introduced by the Belgian mathematical biologist P. F. Verhulst. The approach in this paper is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in partial differential equations.</p>

Diffusive logistic equations with indefinite weights: population models in disrupted environments

1989 ◽  
Vol 112 (3-4) ◽  
pp. 293-318 ◽  
Author(s):  
Robert Stephen Cantrell ◽  
Chris Cosner
Keyword(s):  
Growth Rate ◽  
Diffusion Rate ◽  
Spatial Arrangement ◽  
Lower Solution ◽  
Continuation Methods ◽  
Singular Perturbation Theory ◽  
Local Growth ◽  
Positive Part ◽  
Logistic Equations ◽  
Indefinite Weights

SynopsisThe dynamics of a population inhabiting a strongly heterogeneous environment are modelledby diffusive logistic equations of the form ut = d Δu + [m(x) — cu]u in Ω × (0, ∞), where u represents the population density, c, d > 0 are constants describing the limiting effects of crowding and the diffusion rate of the population, respectively, and m(x) describes the local growth rate of the population. If the environment ∞ is bounded and is surrounded by uninhabitable regions, then u = 0 on ∂∞× (0, ∞). The growth rate m(x) is positive on favourablehabitats and negative on unfavourable ones. The object of the analysis is to determine how the spatial arrangement of favourable and unfavourable habitats affects the population being modelled. The models are shown to possess a unique, stable, positive steady state (implying persistence for the population) provided l/d> where is the principle positive eigenvalue for the problem — Δϕ=λm(x)ϕ in Χ,ϕ=0 on ∂Ω. Analysis of how depends on m indicates that environments with favourable and unfavourable habitats closely intermingled are worse for the population than those containing large regions of uniformly favourable habitat. In the limit as the diffusion rate d ↓ 0, the solutions tend toward the positive part of m(x)/c, and if m is discontinuous develop interior transition layers. The analysis uses bifurcation and continuation methods, the variational characterisation of eigenvalues, upper and lower solution techniques, and singular perturbation theory.

scholarly journals UNIFORMLY-CONVERGENT NUMERICAL METHODS FOR A SYSTEM OF COUPLED SINGULARLY PERTURBED CONVECTION–DIFFUSION EQUATIONS WITH MIXED TYPE BOUNDARY CONDITIONS

2013 ◽  
Vol 18 (5) ◽  
pp. 557-598 ◽  
Author(s):  
Rajarammohanroy Mythili Priyadharshini ◽  
Narashimhan Ramanujam
Keyword(s):  
Boundary Conditions ◽  
Mixed Type ◽  
Coupled System ◽  
Second Order ◽  
Singularly Perturbed ◽  
Supremum Norm ◽  
Convection Diffusion ◽  
Weakly Coupled System ◽  
Type Boundary ◽  
Second Order Convergence

In this paper, two hybrid difference schemes on the Shishkin mesh are constructed for solving a weakly coupled system of two singularly perturbed convection - diffusion second order ordinary differential equations subject to the mixed type boundary conditions. We prove that the method has almost second order convergence in the supremum norm independent of the diffusion parameter. Error bounds for the numerical solution and also the numerical derivative are established. Numerical results are provided to illustrate the theoretical results.

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