scholarly journals Allee dynamics: growth, extinction and range expansion

2017 ◽  
Author(s):  
I Bose ◽  
M Pal ◽  
C Karmakar
Keyword(s):  
Population Density ◽  
Range Expansion ◽  
Population Biology ◽  
Allee Effect ◽  
Additive Noise ◽  
Finite Population ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
One Dimension ◽  
Negative Growth

AbstractIn population biology, the Allee dynamics refer to negative growth rates below a critical population density. In this Letter, we study a reaction-diffusion (RD) model of population growth and dispersion in one dimension, which incorporates the Allee effect in both the growth and mortality rates. In the absence of diffusion, the bifurcation diagram displays regions of both finite population density and zero population density, i.e., extinction. The early signatures of the transition to extinction at a bifurcation point are computed in the presence of additive noise. For the full RD model, the existence of travelling wave solutions of the population density is demonstrated. The parameter regimes in which the travelling wave advances (range expansion) and retreats are identified. In the weak Allee regime, the transition from the pushed to the pulled wave is shown as a function of the mortality rate constant. The results obtained are in agreement with the recent experimental observations on budding yeast populations.

scholarly journals Allee dynamics: Growth, extinction and range expansion

2017 ◽  
Vol 28 (06) ◽  
pp. 1750074 ◽  
Author(s):  
Indrani Bose ◽  
Mainak Pal ◽  
Chiranjit Karmakar
Keyword(s):  
Population Density ◽  
Range Expansion ◽  
Traveling Wave ◽  
Population Biology ◽  
Allee Effect ◽  
Additive Noise ◽  
Finite Population ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Negative Growth

In population biology, the Allee dynamics refer to negative growth rates below a critical population density. In this paper, we study a reaction-diffusion (RD) model of popoulation growth and dispersion in one dimension, which incorporates the Allee effect in both the growth and mortatility rates. In the absence of diffusion, the bifurcation diagram displays regions of both finite population density and zero population density, i.e. extinction. The early signatures of the transition to extinction at the bifurcation point are computed in the presence of additive noise. For the full RD model, the existence of traveling wave solutions of the population density is demonstrated. The parameter regimes in which the traveling wave advances (range expansion) and retreats are identified. In the weak Allee regime, the transition from the pushed to the pulled wave is shown as a function of the mortality rate constant. The results obtained are in agreement with the recent experimental observations on budding yeast populations.

TRAVELLING WAVES IN INVASION PROCESSES WITH PATHOGENS

2008 ◽  
Vol 18 (03) ◽  
pp. 325-349 ◽  
Author(s):  
ARNAUD DUCROT ◽  
MICHEL LANGLAIS
Keyword(s):  
Wave Front ◽  
Allee Effect ◽  
Travelling Waves ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Host Population ◽  
Travelling Wave Solutions ◽  
Front Speed ◽  
Bistable Dynamics ◽  
Si Model

This work is devoted to the study of a singular reaction–diffusion system arising in modelling the introduction of a lethal pathogen within an invading host population. In the absence of the pathogen, the host population exhibits a bistable dynamics (or Allee effect). Earlier numerical simulations of the singular SI model under consideration have exhibited stable travelling waves and also, under some circumstances, a reversal of the wave front speed due to the introduction of the pathogen. Here we prove the existence of such travelling wave solutions, study their linear stability and give analytical conditions yielding a reversal of the wave front speed, i.e. the invading host population may eventually retreat following the introduction of the lethal pathogen.

scholarly journals Propagation direction of the bistable travelling wavefront for delayed non-local reaction diffusion equations

2019 ◽  
Vol 475 (2223) ◽  
pp. 20180898
Author(s):  
Manjun Ma ◽  
Jiajun Yue ◽  
Chunhua Ou
Keyword(s):  
Population Biology ◽  
Wave Speed ◽  
Local Reaction ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Kernel Functions ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Non Local ◽  
Selection Mechanisms

For delayed non-local reaction–diffusion equations arising from population biology, selection mechanisms of the speed sign for the bistable travelling wavefront have not been found. In this paper, based on the theory of asymptotic speeds of spread for monotone semiflows, we firstly provide an interval of values of wave speed and a novel general condition for determining the speed sign by applying the comparison principle and the globally asymptotic stability of the bistable travelling wave. Moreover, through constructing novel upper/lower solutions, we give explicit conditions for the speed sign to be positive or negative. The obtained results are efficiently applied to three classical forms of the kernel functions.

scholarly journals Competitive Reaction-diffusion Systems: Travelling Waves and Numerical Solutions

2019 ◽  
pp. 1-12
Author(s):  
Md. Kamrujjaman ◽  
Asif Ahmed ◽  
Shohel Ahmed
Keyword(s):  
Population Biology ◽  
Numerical Solutions ◽  
Travelling Waves ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Competitive Reaction ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Reaction Diffusion Model ◽  
Non Linear System

In this paper, we consider a competitive reaction-diffusion model to describe the existence of travelling wave solutions of two competing species. Moreover, the non-linear system is also studied by introducing different competitive-cooperative coefficients; constant and spatially distributed which leads to the persistence and extinction of organisms in a heterogeneous environment of population biology. If the diffusion coefficients and other parameters are positive constant, it is seen that one species is in extinction by the other and coexistence is also possible under certain conditions on carrying capacity. The results are numerically investigated by using the Finite difference method (FDM).

scholarly journals The influence of density in population dynamics with strong and weak Allee effect

2021 ◽  
Vol 29 (1) ◽  
Author(s):  
Kamrun Nahar Keya ◽  
Md. Kamrujjaman ◽  
Md. Shafiqul Islam
Keyword(s):  
Population Dynamics ◽  
Allee Effect ◽  
Global Bifurcation ◽  
Reaction Diffusion ◽  
Logistic Growth ◽  
Allee Effects ◽  
Reaction Diffusion Model ◽  
Positive Steady States ◽  
The Stability ◽  
The Impact

AbstractIn this paper, we consider a reaction–diffusion model in population dynamics and study the impact of different types of Allee effects with logistic growth in the heterogeneous closed region. For strong Allee effects, usually, species unconditionally die out and an extinction-survival situation occurs when the effect is weak according to the resource and sparse functions. In particular, we study the impact of the multiplicative Allee effect in classical diffusion when the sparsity is either positive or negative. Negative sparsity implies a weak Allee effect, and the population survives in some domain and diverges otherwise. Positive sparsity gives a strong Allee effect, and the population extinct without any condition. The influence of Allee effects on the existence and persistence of positive steady states as well as global bifurcation diagrams is presented. The method of sub-super solutions is used for analyzing equations. The stability conditions and the region of positive solutions (multiple solutions may exist) are presented. When the diffusion is absent, we consider the model with and without harvesting, which are initial value problems (IVPs) and study the local stability analysis and present bifurcation analysis. We present a number of numerical examples to verify analytical results.

scholarly journals A Laplace transform finite difference scheme for the Fisher-KPP equation

2021 ◽  
Vol 15 ◽  
pp. 174830262199958
Author(s):  
Colin L Defreitas ◽  
Steve J Kane
Keyword(s):  
Finite Difference ◽  
Laplace Transform ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Finite Difference Methods ◽  
Reaction Diffusion ◽  
Numerical Approach ◽  
Travelling Wave ◽  
Reaction Diffusion Equation ◽  
Kpp Equation

This paper proposes a numerical approach to the solution of the Fisher-KPP reaction-diffusion equation in which the space variable is developed using a purely finite difference scheme and the time development is obtained using a hybrid Laplace Transform Finite Difference Method (LTFDM). The travelling wave solutions usually associated with the Fisher-KPP equation are, in general, not deemed suitable for treatment using Fourier or Laplace transform numerical methods. However, we were able to obtain accurate results when some degree of time discretisation is inbuilt into the process. While this means that the advantage of using the Laplace transform to obtain solutions for any time t is not fully exploited, the method does allow for considerably larger time steps than is otherwise possible for finite-difference methods.

scholarly journals Stability and bifurcation in a two-patch model with additive Allee effect

2021 ◽  
Vol 7 (1) ◽  
pp. 536-551
Author(s):  
Lijuan Chen ◽  
◽  
Tingting Liu ◽  
Fengde Chen
Keyword(s):  
Numerical Simulation ◽  
Asymptotic Stability ◽  
Population Density ◽  
Mathematical Analysis ◽  
Allee Effect ◽  
Total Population ◽  
Population Abundance ◽  
Dispersal Rate ◽  
Saddle Node Bifurcation ◽  
Patch Model

<abstract><p>A two-patch model with additive Allee effect is proposed and studied in this paper. Our objective is to investigate how dispersal and additive Allee effect have an impact on the above model's dynamical behaviours. We discuss the local and global asymptotic stability of equilibria and the existence of the saddle-node bifurcation. Complete qualitative analysis on the model demonstrates that dispersal and Allee effect may lead to persistence or extinction in both patches. Also, combining mathematical analysis with numerical simulation, we verify that the total population abundance will increase when the Allee effect constant $ a $ increases or $ m $ decreases. And the total population density increases when the dispersal rate $ D_{1} $ increases or the dispersal rate $ D_{2} $ decreases.</p></abstract>

scholarly journals FKPP equation with impulses on unbounded domain

2017 ◽  
Vol 1 ◽  
pp. 1 ◽  
Author(s):  
Valaire Yatat ◽  
Yves Dumont
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Bare Soil ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Spreading Speed ◽  
Systems Dynamics ◽  
Travelling Wave Solutions ◽  
Minimal Speed ◽  
Wave Solutions

This paper deals with the problem of travelling wave solutions in a scalar impulsive FKPP-like equation. It is a first step of a more general study that aims to address existence of travelling wave solutions for systems of impulsive reaction-diffusion equations that model ecological systems dynamics such as fire-prone savannas. Using results on scalar recursion equations, we show existence of populated vs. extinction travelling waves invasion and compute an explicit expression of their spreading speed (characterized as the minimal speed of such travelling waves). In particular, we find that the spreading speed explicitly depends on the time between two successive impulses. In addition, we carry out a comparison with the case of time-continuous events. We also show that depending on the time between two successive impulses, the spreading speed with pulse events could be lower, equal or greater than the spreading speed in the case of time-continuous events. Finally, we apply our results to a model of fire-prone grasslands and show that pulse fires event may slow down the grassland vs. bare soil invasion speed.

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