Critical Domain Problem for the Reaction–Telegraph Equation Model of Population Dynamics

Alharbi, Weam; Petrovskii, Sergei

doi:10.3390/math6040059

Cited by 21 publications

(14 citation statements)

References 39 publications

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“…In its turn, such attracting nonnegative stationary solution can make the emergence of non-positive solutions less likely. The enhanced nonnegativity of solutions of the nonlinear reaction-telegraph equation was indeed demonstrated numerically in other studies [49,50].…”

Section: Discussionsupporting

confidence: 62%

On the Consistency of the Reaction-Telegraph Process Within Finite Domains

Tilles

Petrovskii

2019

J Stat Phys

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Reaction-telegraph equation (RTE) is a mathematical model that has often been used to describe natural phenomena, with specific applications ranging from physics to social sciences. In particular, in the context of ecology, it is believed to be a more realistic model to describe animal movement than the more traditional approach based on the reaction-diffusion equations. Indeed, the reaction-telegraph equation arises from more realistic microscopic assumptions about individual animal movement (the correlated random walk) and hence could be expected to be more relevant than the diffusion-type models that assume the simple, unbiased Brownian motion. However, the RTE has one significant drawback as its solutions are not positively defined. It is not clear at which stage of the RTE derivation the realism of the microscopic description is lost and/or whether the RTE can somehow be 'improved' to guarantee the solutions positivity. Here we show that the origin of the problem is twofold. Firstly, the RTE is not fully equivalent to the Cattaneo system from which it is obtained; the equivalence can only be achieved in a certain parameter range and only for the initial conditions containing a finite number of Fourier modes. Secondly, the Dirichlet type boundary conditions routinely used for reaction-diffusion equations appear to be meaningless if used for the RTE resulting in solutions with unrealistic properties. We conclude that, for the positivity to be regained, one has to use the Cattaneo system with boundary conditions of Robin type or Neumann type, and we show how relevant classes of solutions can be obtained. Response to Reviewers:Reply to Reviewer 1 All changes made in the manuscript are marked in blue colour.1) The referee is right: there are restrictions on what current fields can be realised within a given density field, and the observed 'population appears from nowhere' feature indeed comes from the imposition of a positive rate of change of density on the

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Section: Discussionsupporting

confidence: 62%

On the Consistency of the Reaction-Telegraph Process Within Finite Domains

Tilles

Petrovskii

2019

J Stat Phys

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“…(4) has a wavelike character with wave speed c = U 0 / √ d (similar to that obtained by Sevilla and Castro-Villarreal [9]), but for times long compared to τ R the behavior is diffusive with the swim diffusivity D swim = U 2 0 τ R /d. Equation (4) in this form is similar to the model created by Alharbi and Petrovskii for population dynamics [15]. Fundamentally, ABP dynamics exhibit both wavelike behavior at short times and diffusive behavior at long times; this behavior is general for all active systems.…”

Section: Theoretical Frameworkmentioning

confidence: 84%

“…This effect is exacerbated by the lack of thermal diffusion, which provides an additional mechanism through which the wavefront can relax, as will be seen in the following section. The telegraph equation shows the essential features of the ballistic to diffusive behavior [15], but it is not sufficient to quantitatively capture the transitional dynamics and is only strictly valid in the limit of high activity.…”

Section: Waves From An Instantaneous Sourcementioning

confidence: 99%

Waves in active matter: The transition from ballistic to diffusive behavior

Dulaney

Brady

2020

Phys. Rev. E

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We highlight the unique wavelike character observed in the relaxation dynamics of active systems via a Smoluchowski based theoretical framework and Brownian dynamic simulations. Persistent swimming motion results in wavelike dynamics until the advective swim displacements become sufficiently uncorrelated, at which point the motion becomes a random walk process characterized by a swim diffusivity, D swim = U 2 0 τ R /[d (d − 1)], dependent on the speed of swimming U 0 , reorientation time τ R , and reorientation dimension d. This change in behavior is described by a telegraph equation, which governs the transition from ballistic wavelike motion to long-time diffusive motion. We study the relaxation of active Brownian particles from an instantaneous source, and provide an explanation for the nonmonotonicity observed in the intermediate scattering function. Using our simple kinetic model we provide the density distribution for the diffusion of active particles released from a line source as a function of time, position, and the ratio of the activity to thermal energy. We extend our analysis to include the effects of an external field on particle spreading to further understand how reorientation events in the active force vector affect relaxation. The strength of the applied external field is shown to be inversely proportional to the decay of the wavelike structure. Our theoretical description for the evolution of the number density agrees with Brownian dynamic simulation data.

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“…For other interesting instances, see Nobile and Ricciardi [16,17] and Román-Román and Torres-Ruiz [18]. Concerning the problem of habitat fragmentation, Alharbi and Petrovskii [19] consider the reaction-telegraph equation as a stochastic model for population growth. They especially focus on the problem of population persistence in small habitats, often called problem of critical domain, and perform a simulation-based analysis concerning the logistic case.…”

Section: Diffusion Processes For Logistic Growthmentioning

confidence: 99%

Logistic Growth Described by Birth-Death and Diffusion Processes

Crescenzo

Paraggio

2019

Mathematics

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We consider the logistic growth model and analyze its relevant properties, such as the limits, the monotony, the concavity, the inflection point, the maximum specific growth rate, the lag time, and the threshold crossing time problem. We also perform a comparison with other growth models, such as the Gompertz, Korf, and modified Korf models. Moreover, we focus on some stochastic counterparts of the logistic model. First, we study a time-inhomogeneous linear birth-death process whose conditional mean satisfies an equation of the same form of the logistic one. We also find a sufficient and necessary condition in order to have a logistic mean even in the presence of an absorbing endpoint. Then, we obtain and analyze similar properties for a simple birth process, too. Then, we investigate useful strategies to obtain two time-homogeneous diffusion processes as the limit of discrete processes governed by stochastic difference equations that approximate the logistic one. We also discuss an interpretation of such processes as diffusion in a suitable potential. In addition, we study also a diffusion process whose conditional mean is a logistic curve. In more detail, for the considered processes we study the conditional moments, certain indices of dispersion, the first-passage-time problem, and some comparisons among the processes.

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Critical Domain Problem for the Reaction–Telegraph Equation Model of Population Dynamics

Cited by 21 publications

References 39 publications

On the Consistency of the Reaction-Telegraph Process Within Finite Domains

On the Consistency of the Reaction-Telegraph Process Within Finite Domains

Waves in active matter: The transition from ballistic to diffusive behavior

Logistic Growth Described by Birth-Death and Diffusion Processes

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