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Modeling with Delay Differential Equations
Published in Sandip Banerjee, Mathematical Modeling, 2021
Allee Effect: It is a general notion from the classical view of population dynamics that the growth rate of the population increases when population size is small and decreases when population size is large (due to intra-specific competition. that is, competition for resources). However, Warder Clyde Allee showed that the reverse is also true in some cases, which he demonstrated for the growth of goldfish in a tank. The “Allee effect” introduced the phenomenon that the growth rate of individuals increases when the population size falls below a certain critical level. A single species delayed population model with the Allee effect was proposed by Gopalswamy and Ladas [51] as dxdt=x(t)a+bx(t−τ)−cx2(t−τ),
Existence of Periodic Solutions for First-Order Difference Equations Subjected to Allee Effects
Published in Hemen Dutta, Mathematical Methods in Engineering and Applied Sciences, 2020
Allee effects refer to a reduction in individual fitness at low population density that can lead to extinction [2,3,5,6,10,16,17,22,24,36,44,46,59]. It is a phenomenon in Biology characterized by a positive interaction between population density and the per-capita population growth rate in small populations. A strong Allee effect is where a population exhibits “Critical size density”, below which the population declines on average and above which it may increase. It is strongly related to the extinction vulnerability of populations. Any ecological mechanism that can lead to a positive relationship between a component of individual fitness and either the number or density of conspecifics can be termed a mechanism of the Allee effect [35,58], depensation [15,20,40], or negative competition effect [60]. A few mechanisms generating Allee effects in species dynamics have been suggested in the literature [6,17]. There are several real-world examples exhibiting the presence of Allee effects [9,16,28,31]. Hence, system analysis considering Allee effects has gained importance in real-world problems in various fields such as population management [6], interacting species [8], biological invasions [13], marine systems [25], conservation biology [27], pest control, biological control [30], sustainable harvesting [41], and meta population dynamics [64]. A critical review of single-species models subject to Allee effects can be found in [7].
Geometrical and Numerical Methods for First-Order Equations
Published in Stephen A. Wirkus, Randall J. Swift, Ryan S. Szypowski, A Course in Differential Equations with Boundary-Value Problems, 2017
Stephen A. Wirkus, Randall J. Swift, Ryan S. Szypowski
An Allee effect model is an improvement over the logistic model in that, if a population gets too small, the growth rate becomes negative. For a small population or a large population, we thus have the per capita growth rate being negative and we can write 1xdxdt=r(a−x)(x−b),
Dynamics analysis and numerical implementations of a diffusive predator–prey model with herd behaviour and Allee effect
Published in Systems Science & Control Engineering, 2022
Now we consider that the prey population is subjected to the Allee effect. The Allee effect mainly signifies a positive relationship between the size of the population and average fitness of the individuals (Berec et al., 2007; Vishwakarma & Sen, 2021; Wang & Kot, 2001). In fact, the reduction of the per capita growth rate of a population of a biological species at densities smaller than a critical value is known as the Allee effect (Hadjiavgousti & Ichtiaroglou, 2008). The main cause of the Allee effect is the difficulty in finding mates between the individuals of a species at low population densities. Other causes may be reduced defense against predators, special trends of social behaviour, etc. In the present work, the model (1) involving the Allee effect given by Wang and Kot (2001) is where is the term for the Allee effect and can be called the ‘Allee effect constant’. The larger α is, the stronger the Allee effect will be, and the slower the per capita growth rate of the prey population: by introducing the Allee effect into the model (1), the per capita growth rate of the prey population is reduced from to .
Global dynamics for a class of reaction–diffusion equations with distributed delay and non-monotone bistable nonlinearity
Published in Applicable Analysis, 2023
Toshikazu Kuniya, Tarik Mohammed Touaoula
In this paper, we are concerned with the bistable case where the equation admits two positive solutions in addition to the trivial solution. In general, the problem (1) in the bistable case is more complicated than that in the monostable case. In this framework, to the best of our knowledge, only little attention has been paid to the dynamics of Equation (1) involving the bistable case. In [13], Huang et al. investigated the following general equation in the bistable case: The authors described the basins of attraction of steady states and obtained a series of invariant intervals using the domain decomposition method. Their results were applied to a population model with Allee effect, illustrated by the functions and . More recently, in [14], Hu and Zhou studied the following model in the bistable case: where . The authors regarded u as the total mature population of a two-stage species and τ as the maturation time for the species. Relating the dynamics of the nonlinear term to stability results for the monostable case, the authors described the basins of attraction of steady states. More precisely, they clarified the existence and size of two equilibria and for each case, and studied the global stability of 0 and and instability of . There results included the existence of a heteroclinic orbit and a periodic orbit.