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

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## Abstract

**:**

## 1. Introduction

## 2. Non-Conservative Property of the Telegraph Equation

## 3. Telegraph Equation with Linear Growth

## 4. Telegraph Equation with Nonlinear Growth

## 5. Empirical Model: Telegraph Equation with a Cutoff

## 6. Discussion and Conclusions

- In the case of a linear reaction term (i.e., linear population growth), we found that the critical domain size for the reaction–telegraph equation coincides with that of the corresponding reaction–diffusion equation. This seems to be a surprising result as intuitively the more directional animal movement described by the reaction–telegraph equation should result in a larger critical size.
- In the case of a nonlinear growth (either logistic or with a strong Allee effect), we found that the critical size of the reaction–telegraph equation is indeed somewhat larger than that of the corresponding reaction–diffusion equation. Thus, the difference between the two models arise as a result of a subtle interplay between the movement pattern and the nonlinearity of the population growth.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Calculation of the Coefficients in the Fourier Series

## Appendix B. Numerical Scheme

## References

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**Figure 1.**Critical domain size of the reaction–telegraph equation obtained numerically at various values of linear growth rate $\alpha $ and various values of ${\omega}_{1}$ and ${\omega}_{2}$ as shown by different symbols and different colors. The solid blue curve corresponds to Equation (28). For parameters from below the curve, the population goes to extinction; for parameters from above the curve, the population exhibits unbounded growth.

**Figure 2.**Dependence of the critical domain size on parameter a α in case the population growth is logistic (see Equation (30)). The dashed black curve shows the results obtained for the reaction–telegraph Equation (29), and the solid red curve shows the results obtained for the corresponding reaction–diffusion equation, i.e., Equation (29) with ω

_{1}= 0. Other parameters are K = 1, D = 1.

**Figure 3.**Dependence of the critical domain size on parameter $\alpha $ in case the population growth is subject to the strong Allee effect (see Equation (31)). The dashed orange curve shows the results for the reaction–telegraph Equation (29), and the solid blue curve shows the results obtained for the corresponding reaction–diffusion equation, i.e., Equation (29) with ${\omega}_{1}=0$. Other parameters are $K=1$, $D=1$, $\beta =0.1$.

**Figure 4.**Solution of the reaction–telegraph equation with the logistic growth without cutoff (

**left**, red curves) and with the cutoff of negative values (

**right**, blue curves). Parameters are ${\omega}_{1}=6$ and ${\omega}_{2}\phantom{\rule{3.33333pt}{0ex}}=2$. The circles mark the location of the areas where the solution becomes negative.

**Figure 5.**Distribution of the error $\parallel u-{u}_{cutoff}\parallel $ over space at $t=0.5$, i.e., immediately after the cutoff was implemented. (

**left**) distribution over the whole domain $0<x<L$; (

**right**) a magnified view of the part of the domain close to the left-hand side boundary.

**Figure 6.**(

**left**) maximum error as a function of time; (

**right**) distribution of error $\parallel u-{u}_{cutoff}\parallel $ over the domain $0<x<L$ at $t=20$.

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Alharbi, W.; Petrovskii, S.
Critical Domain Problem for the Reaction–Telegraph Equation Model of Population Dynamics. *Mathematics* **2018**, *6*, 59.
https://doi.org/10.3390/math6040059

**AMA Style**

Alharbi W, Petrovskii S.
Critical Domain Problem for the Reaction–Telegraph Equation Model of Population Dynamics. *Mathematics*. 2018; 6(4):59.
https://doi.org/10.3390/math6040059

**Chicago/Turabian Style**

Alharbi, Weam, and Sergei Petrovskii.
2018. "Critical Domain Problem for the Reaction–Telegraph Equation Model of Population Dynamics" *Mathematics* 6, no. 4: 59.
https://doi.org/10.3390/math6040059