# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2022077
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## Effects of environmental heterogeneity on species spreading via numerical analysis of some free boundary models

 1 School of Science and Technology, University of New England, Armidale, NSW 2351, Australia 2 Department of Mathematics, Pabna University of Science and Technology, Pabna-6600, Bangladesh

*Corresponding author: Yihong Du

Dedicated to the memory of Professor Masayasu Mimura

Received  September 2021 Revised  January 2022 Early access April 2022

Fund Project: This research was supported by the Australian Research Council and a University of New England PhD scholarship

This paper investigates the effect of environmental heterogeneity on species spreading via numerical simulation of suitable reaction-diffusion models with free boundaries. We focus on the changes of long-time dynamics (establishment or extinction) and spreading speeds of the species as the parameters describing the heterogeneity of the environment are varied. For the single species model in time-periodic environment and in space-periodic environment theoretically treated in [15,16], we obtain more detailed properties here. Among other results, our numerical simulation suggests that, in a time-periodic or space-periodic environment, moderate increase of the oscillation scale enhances the chances of establishment as well as the spreading speed of the species. We also numerically examine a related model with two competing species, which was treated in [34,28,24] recently and reduces to the single species free boundary model when one of the species is absent. Our numerical results, obtained by varying the parameters in the time-periodic and space-periodic terms of the model, suggest that heterogeneity of the environment enhances the invasion of the two species (as in the single species model), although there are subtle differences of the influences felt by the two. Some intriguing phenomena revealed in our simulations suggest that heterogeneity of the environment decreases the level of predictability of the competition outcome.

Citation: Kamruzzaman Khan, Timothy M. Schaerf, Yihong Du. Effects of environmental heterogeneity on species spreading via numerical analysis of some free boundary models. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022077
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##### References:
The graph of the population density function $u(x,t)$ at time $t = 120$ (solid green curve) almost coincides with that of $u(x,t)$ at time $t = 121$ (red dotted curve), indicating time-periodic variation of the population induced by the time-periodic environment during the spreading process
Evolution of the population density $u(x,t)$ up to time $t = 120$ in the simulated time-periodic environment. Note that in the graph, $u(x,t)$ was extended to 0 for $x>s(t)$
Graph of spreading front function $s(t)$
Spreading speed as a function of period $T$ in a time-periodic environment
Profiles of $u(x,t)$ in a space-periodic environment with period $L = 1$ and magnitude of oscillation $\sigma = 1$. Snapshots are taken at times $t = 84.8$, $t = 102.4$, and $t = 120$, indicating space-periodic variation of the population induced by the space-periodic environment during the spreading process
Evolution of the population density function $u(x,t)$ up to time $t = 120$ in the simulated space-periodic environment. Note that $u(x,t)$ was extended to 0 for $x>s(t)$
Corresponding spreading front function $s(t)$ up to time $t = 120$
The blue solid line represents the speed of the expanding front at $t = 120$ in a homogeneous environment, the green solid line depicts the average speed of the expanding front at $t = 120$ in the simulated time-periodic environment and the red solid line stands for the average speed of the expanding front at $t = 120$ in the simulated space-periodic environment
Spreading speed as a function of period $L$ in a space-periodic environment
Evolution of the population density functions $u(x,t)$ and $v(x,t)$ up to time $t = 120$ in the simulated time-periodic environment. Note that the extensions $u(x,t) = 0$ for $x>s_1(t)$ and $v(x, t) = 0$ for $x>s_2(t)$ were used in the graph
Spreading fronts of the species up to time $t = 120$
Average speeds of $u$ and $v$ with $\sigma_1 = \sigma_2 \in [0.1,2]$: The average speed of $u$ at $t = 120$ (i.e., $\overline{s_1'(120)}$) (the green curve in left figure) increases while the average speed of $v$ at $t = 120$ (i.e., $\overline{s_2'(120)}$ (magenta line in right figure) decreases as $\sigma_1 = \sigma_2$ increases in $[0.1,2]$. Compared with the corresponding speed of $u$ and $v$ in a homogeneous environment (the blue and red lines in the left and right figures, respectively), the speed of $u$ is slightly bigger in the time-periodic environment, while the speed of $v$ is slightly smaller
Average speeds of $u$ and $v$ with $\sigma_1 = 1$ and $\sigma_2 \in [0.1,2]$: The average speed of $u$ at $t = 120$ (i.e., $\overline{s_1'(120)}$) decreases very slowly (green curve in left figure) as $\sigma_2$ is increased in $[0.1,2]$, but it is bigger than the corresponding speed in the homogeneous environment (blue line); the average speed of $v$ at $t = 120$ (i.e., $\overline{s_2'(120)}$) also decreases (magenta curve in right figure) as $\sigma_2$ increases in [0.1, 2], but in contrast, it is smaller than the speed of $v$ in homogeneous environment (red line)
Average speeds of $u$ and $v$ with $\sigma_2 = 1$ and $\sigma_1 \in [0.1,2]$: The average speed of $u$ at $t = 120$ (i.e., $\overline{s_1'(120)}$) increases (green curve in left figure) when $\sigma_1$ increases in $[0.1,2]$, and it is bigger than the corresponding speed of $u$ in a homogeneous environment (i.e., $\sigma_1 = \sigma_2 = 0$); while the average speed of $v$ at $t = 120$ (i.e., $\overline{s_2'(120)}$) stays constant (magenta line in right figure) as $\sigma_1$ varies in $[0.1, 2]$, and in contrast, it is smaller than the corresponding speed of $v$ (i.e., $s_2'(120)$) in a homogeneous environment
Moving fronts of $u$ and $v$ for a selection of values of $T$ in Table 3.8
Spreading speeds of $u$ and $v$ for a selection of values of $T$ in Table 3.8
Population distribution of $u$ and $v$ at time $t = 10$ with $T = 0.6, 1.0, 1.1, 1.5, 1.8,1.9, 10$
Population distribution of $u$ and $v$ at time $t = 0, 10, 30, 50,120$ with $T = 0.6, 1.5, 2, 10$
Profiles of $u(x,t)$ and $v(x,t)$ in a space-periodic environment of period $L = 1$ and magnitude of the oscillations $\sigma_1 = \sigma_2 = 1$
Evolution of the population density functions $u(x,t)$ and $v(x,t)$ up to time $t = 120$ in the simulated space-periodic environment. The extensions $u(x,t) = 0$ for $x>s_1(t)$ and $v(x,t) = 0$ for $x>s_2(t)$ were used in the graph
Corresponding spreading front functions of the species in Figure 3.19
Spreading speeds of $u$ (left) and $v$ (right) in homogeneous and space-periodic environments with period $L = 1$ and magnitude of the oscillations $\sigma_1 = 1.0$, $\sigma_2 = 1.0$
Average speeds of $u$ and $v$ with $\sigma_1 = \sigma_2 \in [0.1,2]$: The average speeds of $u$ (green curve in left figure) and $v$ (magenta curve in right figure) both increase as $\sigma_1 = \sigma_2$ are increased. They are both bigger than the corresponding speeds in a homogeneous environment
Average speeds of $u$ and $v$ with $\sigma_1 = 1$ and $\sigma_2 \in [0.1,2]$: The average speed of $u$ decreases (green curve in left figure) when $\sigma_2$ is increased in $[0.1,2]$, while the average speed of $v$ increases with $\sigma_2$ (magenta curve in right figure). Both speeds are greater than the corresponding speeds in a homogeneous environment
Average speeds of $u$ and $v$ with $\sigma_2 = 1$ and $\sigma_1 \in [0.1,2]$: The average speed of $u$ increases (green curve in left figure) when $\sigma_1$ is increased in $[0.1,2]$, while the average speed of $v$ does not change with $\sigma_1$ (magenta line in right figure). Both speeds are greater than the corresponding speeds in a homogeneous environment except that of $u$ for $\sigma_1\leq 0.4$
Changes of spreading speeds as $L$ varies given in Table 3.9
Moving fronts of $u$ and $v$ for a selection of values of $L$ in Table 3.9
Spreading speeds of $u$ and $v$ for a selection of values of $L$ in Table 3.9
Moving fronts of $u$ and $v$ with $L = 5, 10$
Spreading speeds of $u$ and $v$ with $L = 5, 10$
Population distribution of $u$ and $v$ at time $t = 0, 10, 40, 80,120$ with $L = 1, 2, 3, 5$
Spreading range of $\lambda$ with time period $T = 1$ and changing magnitude $\sigma$
 Oscillation magnitude $\sigma$ Spreading range of $\lambda$ 0.2 $(0.445, \infty)$ 0.4 $(0.445, \infty)$ 0.6 $(0.445, \infty)$ 0.8 $(0.435, \infty)$ 1.0 $(0.435, \infty)$ 1.5 $(0.435, \infty)$ 2.0 $(0.435, \infty)$ 5.0 $(0.405, \infty)$
 Oscillation magnitude $\sigma$ Spreading range of $\lambda$ 0.2 $(0.445, \infty)$ 0.4 $(0.445, \infty)$ 0.6 $(0.445, \infty)$ 0.8 $(0.435, \infty)$ 1.0 $(0.435, \infty)$ 1.5 $(0.435, \infty)$ 2.0 $(0.435, \infty)$ 5.0 $(0.405, \infty)$
Spreading range of $\lambda$ with $\sigma = 1$ and varying time period $T$
 Time period $T$ Spreading range of $\lambda$ 0.2 $(0.445, \infty)$ 0.4 $(0.445, \infty)$ 0.6 $(0.445, \infty)$ 0.8 $(0.435, \infty)$ 1.0 $(0.435, \infty)$ 2.0 $(0.435, \infty)$ 5.0 $(0.415, \infty)$ 10.0 $(0.385, \infty)$ 20.0 $(0.365, \infty)$
 Time period $T$ Spreading range of $\lambda$ 0.2 $(0.445, \infty)$ 0.4 $(0.445, \infty)$ 0.6 $(0.445, \infty)$ 0.8 $(0.435, \infty)$ 1.0 $(0.435, \infty)$ 2.0 $(0.435, \infty)$ 5.0 $(0.415, \infty)$ 10.0 $(0.385, \infty)$ 20.0 $(0.365, \infty)$
Spreading range of $\lambda$ with space-period $L = 1$ and varying $\sigma$
 Oscillation magnitude $\sigma$ Spreading range of $\lambda$ 0.2 $(0.435, \infty)$ 0.4 $(0.425, \infty)$ 0.6 $(0.415, \infty)$ 0.8 $(0.405, \infty)$ 1.0 $(0.395, \infty)$ 1.5 $(0.365, \infty)$ 2.0 $(0.335, \infty)$ 5.0 $(0.215, \infty)$
 Oscillation magnitude $\sigma$ Spreading range of $\lambda$ 0.2 $(0.435, \infty)$ 0.4 $(0.425, \infty)$ 0.6 $(0.415, \infty)$ 0.8 $(0.405, \infty)$ 1.0 $(0.395, \infty)$ 1.5 $(0.365, \infty)$ 2.0 $(0.335, \infty)$ 5.0 $(0.215, \infty)$
Spreading range of $\lambda$ with $\sigma = 1$ and varying $L$
 Period $L$ Spreading range of $\lambda$ 0.2 $(0.435, \infty)$ 0.4 $(0.425, \infty)$ 0.6 $(0.415, \infty)$ 0.8 $(0.405, \infty)$ 1.0 $(0.395, \infty)$ 2.0 $(0.335, \infty)$ 5.0 $(0.365, \infty)$ 10.0 $(0.395, \infty)$ 20.0 $(0.415, \infty)$
 Period $L$ Spreading range of $\lambda$ 0.2 $(0.435, \infty)$ 0.4 $(0.425, \infty)$ 0.6 $(0.415, \infty)$ 0.8 $(0.405, \infty)$ 1.0 $(0.395, \infty)$ 2.0 $(0.335, \infty)$ 5.0 $(0.365, \infty)$ 10.0 $(0.395, \infty)$ 20.0 $(0.415, \infty)$
Change of long-time behaviour as the initial functions are varied in homogeneous, time-periodic and space-periodic environments - A
 $\lambda_1$ Environments Both $u$ and $v$ vanishing Only $v$ vanishing Chase-and-run coexistence Only $u$ vanishing 0.10 Homogeneous: $\lambda_2\le 0.223$ NA NA $\lambda_2\ge 0.224$ Time-Periodic: $\lambda_2\le 0.212$ NA NA $\lambda_2\ge 0.213$ Space-Periodic: $\lambda_2\le 0.190$ NA NA $\lambda_2\ge 0.191$ 0.50 Homogeneous NA $\lambda_2 \le 0.367$ $0.368 \le \lambda_2 \le 0.375$ $\lambda_2\ge 0.376$ Time-Periodic: NA $\lambda_2 \le 0.380$ $0.381 \le \lambda_2 \le 0.390$ $\lambda_2\ge 0.391$ Space-Periodic: NA $\lambda_2 \le 0.444$ $0.445 \le \lambda_2 \le 0.583$ $\lambda_2 \ge 0.584$ 0.75 Homogeneous: NA $\lambda_2\le 0.840$ $\lambda_2\ge 0.841$ NA Time-Periodic: NA $\lambda_2\le 0.847$ $\lambda_2\ge 0.848$ NA Space-Periodic: NA $\lambda_2\le 0.887$ $\lambda_2\ge 0.888$ NA 1.0 Homogeneous: NA $\lambda_2\le 1.151$ $\lambda_2\ge 1.152$ NA Time-Periodic: NA $\lambda_2\le 1.152$ $\lambda_2\ge 1.153$ NA Space-Periodic: NA $\lambda_2\le 1.168$ $\lambda_2\ge 1.169$ NA 2.0 Homogeneous: NA $\lambda_2\le 2.184$ $\lambda_2\ge 2.185$ NA Time-Periodic: NA $\lambda_2\le 2.186$ $\lambda_2\ge 2.187$ NA Space-Periodic: NA $\lambda_2\le 2.182$ $\lambda_2\ge 2.183$ NA 3.0 Homogeneous: NA $\lambda_2\le 3.204$ $\lambda_2\ge 3.205$ NA Time-Periodic: NA $\lambda_2\le 3.206$ $\lambda_2\ge 3.207$ NA Space-Periodic: NA $\lambda_2\le 3.202$ $\lambda_2\ge 3.203$ NA 4.0 Homogeneous: NA $\lambda_2\le 4.220$ $\lambda_2\ge 4.221$ NA Time-Periodic: NA $\lambda_2\le 4.223$ $\lambda_2\ge 4.224$ NA Space-Periodic: NA $\lambda_2\le 4.2202$ $\lambda_2\ge 4.2203$ NA 5.0 Homogeneous: NA $\lambda_2\le 5.234$ $\lambda_2\ge 5.235$ NA Time-Periodic: NA $\lambda_2\le 5.237$ $\lambda_2\ge 5.238$ NA Space-Periodic: NA $\lambda_2\le 5.233$ $\lambda_2\ge 5.234$ NA 6.0 Homogeneous: NA $\lambda_2\le 6.246$ $\lambda_2\ge 6.247$ NA Time-Periodic: NA $\lambda_2\le 6.248$ $\lambda_2\ge 6.249$ NA Space-Periodic: NA $\lambda_2\le 6.244$ $\lambda_2\ge 6.245$ NA 7.0 Homogeneous: NA $\lambda_2\le 7.256$ $\lambda_2\ge 7.257$ NA Time-Periodic: NA $\lambda_2\le 7.258$ $\lambda_2\ge 7.259$ NA Space-Periodic: NA $\lambda_2\le 7.253$ $\lambda_2\ge 7.254$ NA 8.0 Homogeneous: NA $\lambda_2\le 8.264$ $\lambda_2\ge 8.265$ NA Time-Periodic: NA $\lambda_2\le 8.267$ $\lambda_2\ge 8.268$ NA Space-Periodic: NA $\lambda_2\le 8.262$ $\lambda_2\ge 8.263$ NA 9.0 Homogeneous: NA $\lambda_2\le 9.272$ $\lambda_2\ge 9.273$ NA Time-Periodic: NA $\lambda_2\le 9.274$ $\lambda_2\ge 9.275$ NA Space-Periodic: NA $\lambda_2\le 9.269$ $\lambda_2\ge 9.270$ NA
 $\lambda_1$ Environments Both $u$ and $v$ vanishing Only $v$ vanishing Chase-and-run coexistence Only $u$ vanishing 0.10 Homogeneous: $\lambda_2\le 0.223$ NA NA $\lambda_2\ge 0.224$ Time-Periodic: $\lambda_2\le 0.212$ NA NA $\lambda_2\ge 0.213$ Space-Periodic: $\lambda_2\le 0.190$ NA NA $\lambda_2\ge 0.191$ 0.50 Homogeneous NA $\lambda_2 \le 0.367$ $0.368 \le \lambda_2 \le 0.375$ $\lambda_2\ge 0.376$ Time-Periodic: NA $\lambda_2 \le 0.380$ $0.381 \le \lambda_2 \le 0.390$ $\lambda_2\ge 0.391$ Space-Periodic: NA $\lambda_2 \le 0.444$ $0.445 \le \lambda_2 \le 0.583$ $\lambda_2 \ge 0.584$ 0.75 Homogeneous: NA $\lambda_2\le 0.840$ $\lambda_2\ge 0.841$ NA Time-Periodic: NA $\lambda_2\le 0.847$ $\lambda_2\ge 0.848$ NA Space-Periodic: NA $\lambda_2\le 0.887$ $\lambda_2\ge 0.888$ NA 1.0 Homogeneous: NA $\lambda_2\le 1.151$ $\lambda_2\ge 1.152$ NA Time-Periodic: NA $\lambda_2\le 1.152$ $\lambda_2\ge 1.153$ NA Space-Periodic: NA $\lambda_2\le 1.168$ $\lambda_2\ge 1.169$ NA 2.0 Homogeneous: NA $\lambda_2\le 2.184$ $\lambda_2\ge 2.185$ NA Time-Periodic: NA $\lambda_2\le 2.186$ $\lambda_2\ge 2.187$ NA Space-Periodic: NA $\lambda_2\le 2.182$ $\lambda_2\ge 2.183$ NA 3.0 Homogeneous: NA $\lambda_2\le 3.204$ $\lambda_2\ge 3.205$ NA Time-Periodic: NA $\lambda_2\le 3.206$ $\lambda_2\ge 3.207$ NA Space-Periodic: NA $\lambda_2\le 3.202$ $\lambda_2\ge 3.203$ NA 4.0 Homogeneous: NA $\lambda_2\le 4.220$ $\lambda_2\ge 4.221$ NA Time-Periodic: NA $\lambda_2\le 4.223$ $\lambda_2\ge 4.224$ NA Space-Periodic: NA $\lambda_2\le 4.2202$ $\lambda_2\ge 4.2203$ NA 5.0 Homogeneous: NA $\lambda_2\le 5.234$ $\lambda_2\ge 5.235$ NA Time-Periodic: NA $\lambda_2\le 5.237$ $\lambda_2\ge 5.238$ NA Space-Periodic: NA $\lambda_2\le 5.233$ $\lambda_2\ge 5.234$ NA 6.0 Homogeneous: NA $\lambda_2\le 6.246$ $\lambda_2\ge 6.247$ NA Time-Periodic: NA $\lambda_2\le 6.248$ $\lambda_2\ge 6.249$ NA Space-Periodic: NA $\lambda_2\le 6.244$ $\lambda_2\ge 6.245$ NA 7.0 Homogeneous: NA $\lambda_2\le 7.256$ $\lambda_2\ge 7.257$ NA Time-Periodic: NA $\lambda_2\le 7.258$ $\lambda_2\ge 7.259$ NA Space-Periodic: NA $\lambda_2\le 7.253$ $\lambda_2\ge 7.254$ NA 8.0 Homogeneous: NA $\lambda_2\le 8.264$ $\lambda_2\ge 8.265$ NA Time-Periodic: NA $\lambda_2\le 8.267$ $\lambda_2\ge 8.268$ NA Space-Periodic: NA $\lambda_2\le 8.262$ $\lambda_2\ge 8.263$ NA 9.0 Homogeneous: NA $\lambda_2\le 9.272$ $\lambda_2\ge 9.273$ NA Time-Periodic: NA $\lambda_2\le 9.274$ $\lambda_2\ge 9.275$ NA Space-Periodic: NA $\lambda_2\le 9.269$ $\lambda_2\ge 9.270$ NA
Change of long-time behaviour as the initial functions are varied in homogeneous, time-periodic and space-periodic environments - B
 $\lambda_2$ Environments Both $u$ and $v$ vanishing Only $u$ vanishing Chase-and-run coexistence Only $v$ vanishing 0.10 Homogeneous: $\lambda_1\le 0.444$ NA NA $\lambda_1\ge 0.445$ Time-Periodic: $\lambda_1\le 0.438$ NA NA $\lambda_1\ge 0.439$ Space-Periodic: $\lambda_1\le 0.393$ NA NA $\lambda_1\ge 0.394$ 0.50 Homogeneous: NA $\lambda_1\le 0.541$ $0.542 \le \lambda_1 \le 0.564$ $\lambda_1 \ge 0.565$ Time-Periodic: NA $\lambda_1\le 0.535$ $0.536 \le \lambda_1 \le 0.557$ $\lambda_1\ge 0.558$ Space-Periodic: NA $\lambda_1 \le 0.487$ $0.488 \le \lambda_1 \le 0.523$ $\lambda_1 \ge 0.524$ 0.75 Homogeneous: NA $\lambda_1\le 0.585$ $0.586 \le \lambda_1 \le 0.692$ $\lambda_1 \ge 0.693$ Time-Periodic: NA $\lambda_1\le 0.579$ $0.580 \le \lambda_1 \le 0.686$ $\lambda_1\ge 0.687$ Space-Periodic: NA $\lambda_1 \le 0.516$ $0.517 \le \lambda_1 \le 0.652$ $\lambda_1 \ge 0.653$ 1.0 Homogeneous: NA $\lambda_1\le 0.602$ $0.603 \le \lambda_1 \le 0.869$ $\lambda_1 \ge 0.870$ Time-Periodic: NA $\lambda_1\le 0.597$ $0.598 \le \lambda_1 \le 0.866$ $\lambda_1\ge 0.867$ Space-Periodic: NA $\lambda_1 \le 0.528$ $0.529 \le \lambda_1 \le 0.839$ $\lambda_1 \ge 0.840$ 2.0 Homogeneous: NA $\lambda_1\le 0.611$ $0.612 \le \lambda_1 \le 1.819$ $\lambda_1 \ge 1.820$ Time-Periodic: NA $\lambda_1\le 0.606$ $0.607 \le \lambda_1 \le 1.817$ $\lambda_1\ge 1.818$ Space-Periodic: NA $\lambda_1 \le 0.533$ $0.534 \le \lambda_1 \le 1.817$ $\lambda_1 \ge 1.818$ 3.0 Homogeneous: NA $\lambda_1\le 0.609$ $0.610 \le \lambda_1 \le 2.799$ $\lambda_1 \ge 2.800$ Time-Periodic: NA $\lambda_1\le 0.603$ $0.604 \le \lambda_1 \le 2.796$ $\lambda_1\ge 2.797$ Space-Periodic: NA $\lambda_1 \le 0.530$ $0.531 \le \lambda_1 \le 2.800$ $\lambda_1 \ge 2.801$ 4.0 Homogeneous: NA $\lambda_1\le 0.606$ $0.607 \le \lambda_1 \le 3.782$ $\lambda_1 \ge 3.783$ Time-Periodic: NA $\lambda_1\le 0.601$ $0.602 \le \lambda_1 \le 3.779$ $\lambda_1\ge 3.780$ Space-Periodic: NA $\lambda_1 \le 0.527$ $0.528 \le \lambda_1 \le 3.784$ $\lambda_1 \ge 3.785$ 5.0 Homogeneous: NA $\lambda_1\le 0.604$ $0.605 \le \lambda_1 \le 4.768$ $\lambda_1 \ge 4.769$ Time-Periodic: NA $\lambda_1\le 0.599$ $0.600 \le \lambda_1 \le 4.765$ $\lambda_1\ge 4.766$ Space-Periodic: NA $\lambda_1 \le 0.525$ $0.526 \le \lambda_1 \le 4.770$ $\lambda_1 \ge 4.771$
 $\lambda_2$ Environments Both $u$ and $v$ vanishing Only $u$ vanishing Chase-and-run coexistence Only $v$ vanishing 0.10 Homogeneous: $\lambda_1\le 0.444$ NA NA $\lambda_1\ge 0.445$ Time-Periodic: $\lambda_1\le 0.438$ NA NA $\lambda_1\ge 0.439$ Space-Periodic: $\lambda_1\le 0.393$ NA NA $\lambda_1\ge 0.394$ 0.50 Homogeneous: NA $\lambda_1\le 0.541$ $0.542 \le \lambda_1 \le 0.564$ $\lambda_1 \ge 0.565$ Time-Periodic: NA $\lambda_1\le 0.535$ $0.536 \le \lambda_1 \le 0.557$ $\lambda_1\ge 0.558$ Space-Periodic: NA $\lambda_1 \le 0.487$ $0.488 \le \lambda_1 \le 0.523$ $\lambda_1 \ge 0.524$ 0.75 Homogeneous: NA $\lambda_1\le 0.585$ $0.586 \le \lambda_1 \le 0.692$ $\lambda_1 \ge 0.693$ Time-Periodic: NA $\lambda_1\le 0.579$ $0.580 \le \lambda_1 \le 0.686$ $\lambda_1\ge 0.687$ Space-Periodic: NA $\lambda_1 \le 0.516$ $0.517 \le \lambda_1 \le 0.652$ $\lambda_1 \ge 0.653$ 1.0 Homogeneous: NA $\lambda_1\le 0.602$ $0.603 \le \lambda_1 \le 0.869$ $\lambda_1 \ge 0.870$ Time-Periodic: NA $\lambda_1\le 0.597$ $0.598 \le \lambda_1 \le 0.866$ $\lambda_1\ge 0.867$ Space-Periodic: NA $\lambda_1 \le 0.528$ $0.529 \le \lambda_1 \le 0.839$ $\lambda_1 \ge 0.840$ 2.0 Homogeneous: NA $\lambda_1\le 0.611$ $0.612 \le \lambda_1 \le 1.819$ $\lambda_1 \ge 1.820$ Time-Periodic: NA $\lambda_1\le 0.606$ $0.607 \le \lambda_1 \le 1.817$ $\lambda_1\ge 1.818$ Space-Periodic: NA $\lambda_1 \le 0.533$ $0.534 \le \lambda_1 \le 1.817$ $\lambda_1 \ge 1.818$ 3.0 Homogeneous: NA $\lambda_1\le 0.609$ $0.610 \le \lambda_1 \le 2.799$ $\lambda_1 \ge 2.800$ Time-Periodic: NA $\lambda_1\le 0.603$ $0.604 \le \lambda_1 \le 2.796$ $\lambda_1\ge 2.797$ Space-Periodic: NA $\lambda_1 \le 0.530$ $0.531 \le \lambda_1 \le 2.800$ $\lambda_1 \ge 2.801$ 4.0 Homogeneous: NA $\lambda_1\le 0.606$ $0.607 \le \lambda_1 \le 3.782$ $\lambda_1 \ge 3.783$ Time-Periodic: NA $\lambda_1\le 0.601$ $0.602 \le \lambda_1 \le 3.779$ $\lambda_1\ge 3.780$ Space-Periodic: NA $\lambda_1 \le 0.527$ $0.528 \le \lambda_1 \le 3.784$ $\lambda_1 \ge 3.785$ 5.0 Homogeneous: NA $\lambda_1\le 0.604$ $0.605 \le \lambda_1 \le 4.768$ $\lambda_1 \ge 4.769$ Time-Periodic: NA $\lambda_1\le 0.599$ $0.600 \le \lambda_1 \le 4.765$ $\lambda_1\ge 4.766$ Space-Periodic: NA $\lambda_1 \le 0.525$ $0.526 \le \lambda_1 \le 4.770$ $\lambda_1 \ge 4.771$
From coexistence to vanishing of $v$ due to time-periodic variation ($T = 1, \; \sigma_1 = \sigma_2 = 1$)
 $\lambda_1$ $\lambda_2$ Time-Periodic Case Homogeneous Case $1$ $1.152$ Vanishing of $v$ Chase-and-run coexistence $2$ $2.185$ $3$ $3.205$ $4$ $4.223$ $5$ $5.235$ $6$ $6.247$ $7$ $7.257$ $10$ $10.280$
 $\lambda_1$ $\lambda_2$ Time-Periodic Case Homogeneous Case $1$ $1.152$ Vanishing of $v$ Chase-and-run coexistence $2$ $2.185$ $3$ $3.205$ $4$ $4.223$ $5$ $5.235$ $6$ $6.247$ $7$ $7.257$ $10$ $10.280$
Change of long-time behaviour in time-periodic environment as $\sigma_1$ and $\sigma_2$ vary, with a selection of initial data
 $\lambda_2$ $\lambda_1$ $\sigma_1$ $\sigma_2$ Time-Periodic Case Homogeneous Case 1 0.602 1 1 chase-and-run coexistence vanishing of $u$ 0.1 1 vanishing of $u$ 0.869 1 1 vanishing of $v$ chase-and-run coexistence 0.01 0.01 chase-and-run coexistence 2 0.611 1 1 chase-and-run coexistence vanishing of $u$ 0.1 1 vanishing of $u$ 1.819 1 1 vanishing of $v$ chase-and-run coexistence 0.01 0.01 chase-and-run coexistence 3 0.609 1 1 chase-and-run coexistence vanishing of $u$ 0.1 1 vanishing of $u$ 2.799 1 1 vanishing of $v$ chase-and-run coexistence 0.01 0.01 chase-and-run coexistence 4 0.606 1 1 chase-and-run coexistence vanishing of $u$ 0.1 1 vanishing of $u$ 3.782 1 1 vanishing of $v$ chase-and-run coexistence 0.01 0.01 chase-and-run coexistence 5 0.604 1 1 chase-and-run coexistence vanishing of $u$ 0.1 1 vanishing of $u$ 4.768 1 1 vanishing of $v$ chase-and-run coexistence 0.01 0.01 chase-and-run coexistence
 $\lambda_2$ $\lambda_1$ $\sigma_1$ $\sigma_2$ Time-Periodic Case Homogeneous Case 1 0.602 1 1 chase-and-run coexistence vanishing of $u$ 0.1 1 vanishing of $u$ 0.869 1 1 vanishing of $v$ chase-and-run coexistence 0.01 0.01 chase-and-run coexistence 2 0.611 1 1 chase-and-run coexistence vanishing of $u$ 0.1 1 vanishing of $u$ 1.819 1 1 vanishing of $v$ chase-and-run coexistence 0.01 0.01 chase-and-run coexistence 3 0.609 1 1 chase-and-run coexistence vanishing of $u$ 0.1 1 vanishing of $u$ 2.799 1 1 vanishing of $v$ chase-and-run coexistence 0.01 0.01 chase-and-run coexistence 4 0.606 1 1 chase-and-run coexistence vanishing of $u$ 0.1 1 vanishing of $u$ 3.782 1 1 vanishing of $v$ chase-and-run coexistence 0.01 0.01 chase-and-run coexistence 5 0.604 1 1 chase-and-run coexistence vanishing of $u$ 0.1 1 vanishing of $u$ 4.768 1 1 vanishing of $v$ chase-and-run coexistence 0.01 0.01 chase-and-run coexistence
Change of long-time behaviour in space-periodic environment as $\sigma_1$ and $\sigma_2$ vary, with a selection of initial data
 $\lambda_1$ $\lambda_2$ $\sigma_1$ $\sigma_2$ Space-Periodic Case Homogeneous Case 1 1.168 1 1 vanishing of $v$ chase-and-run coexistence 2 chase-and-run coexistence 2 2.182 1 1 vanishing of $v$ vanishing of $v$ 2 chase-and-run coexistence 3 3.202 1 1 vanishing of $v$ vanishing of $v$ 2 chase-and-run coexistence 4 4.2202 1 1 vanishing of $v$ vanishing of $v$ 2 chase-and-run coexistence 5 5.233 1 1 vanishing of $v$ vanishing of $v$ 2 chase-and-run coexistence 6 6.244 1 1 vanishing of $v$ vanishing of $v$ 2 chase-and-run coexistence 7 7.253 1 1 vanishing of $v$ vanishing of $v$ 2 chase-and-run coexistence 10 10.276 1 1 vanishing of $v$ vanishing of $v$ 2 chase-and-run coexistence
 $\lambda_1$ $\lambda_2$ $\sigma_1$ $\sigma_2$ Space-Periodic Case Homogeneous Case 1 1.168 1 1 vanishing of $v$ chase-and-run coexistence 2 chase-and-run coexistence 2 2.182 1 1 vanishing of $v$ vanishing of $v$ 2 chase-and-run coexistence 3 3.202 1 1 vanishing of $v$ vanishing of $v$ 2 chase-and-run coexistence 4 4.2202 1 1 vanishing of $v$ vanishing of $v$ 2 chase-and-run coexistence 5 5.233 1 1 vanishing of $v$ vanishing of $v$ 2 chase-and-run coexistence 6 6.244 1 1 vanishing of $v$ vanishing of $v$ 2 chase-and-run coexistence 7 7.253 1 1 vanishing of $v$ vanishing of $v$ 2 chase-and-run coexistence 10 10.276 1 1 vanishing of $v$ vanishing of $v$ 2 chase-and-run coexistence
Change of long-time behaviour in a time-periodic environment when the time period $T$ is varied while $\sigma_1 = \sigma_2 = 1$ and $\lambda_1 = 1$, $\lambda_2 = 1.153$
 $T$ Time-Periodic Case Homogeneous Case $0.2$ Chase-and-run coexistence Chase-and-run coexistence $0.4$ $0.6$ $0.8$ $1.0$ $1.1$ Vanishing of $v$ $1.5$ $1.8$ $1.9$ Chase-and-run coexistence $2$ $5$ $10$ $15$
 $T$ Time-Periodic Case Homogeneous Case $0.2$ Chase-and-run coexistence Chase-and-run coexistence $0.4$ $0.6$ $0.8$ $1.0$ $1.1$ Vanishing of $v$ $1.5$ $1.8$ $1.9$ Chase-and-run coexistence $2$ $5$ $10$ $15$
Change of long-time behaviour in space-periodic environment when the space period $L$ is varied while $\sigma_1 = \sigma_2 = 1$ and $\lambda_1 = 1$, $\lambda_2 = 1.169$
 $L$ Space-Periodic Case Homogeneous Case $0.2$ Chase-and-run coexistence Chase-and-run coexistence $0.4$ $0.6$ $0.8$ $1.0$ $1.1$ $1.5$ $1.6$ Vanishing of $v$ $1.8$ $2$ $2.2$ $2.3$ Chase-and-run coexistence $2.5$ $2.8$ $3.0$ $3.2$ $3.3$ Vanishing of $v$ $5$ $10$ $15$ $20$
 $L$ Space-Periodic Case Homogeneous Case $0.2$ Chase-and-run coexistence Chase-and-run coexistence $0.4$ $0.6$ $0.8$ $1.0$ $1.1$ $1.5$ $1.6$ Vanishing of $v$ $1.8$ $2$ $2.2$ $2.3$ Chase-and-run coexistence $2.5$ $2.8$ $3.0$ $3.2$ $3.3$ Vanishing of $v$ $5$ $10$ $15$ $20$
Average spreading speeds of $u$ and $v$ at time $t = 120$ with $\sigma_1 = \sigma_2 = 1$ and different values of $T$
 $T$ Time-Periodic $\overline{s_1'(120)}$ Time-Periodic $\overline{s'_2(120)}$ $0.2$ 0.045093 0.364358 $0.4$ 0.045099 0.364336 $0.6$ 0.045118 0.364294 $0.8$ 0.045158 0.364233 $1.0$ 0.045224 0.364154 $1.1$ 0.056211 0 $1.2$ 0.056282 0 $1.4$ 0.056449 0 $1.5$ 0.056545 0 $1.6$ 0.056648 0 $1.8$ 0.056874 0 $1.9$ 0.045852 0.363676 $2$ 0.045949 0.363617 $5$ 0.048875 0.362510 $10$ 0.050970 0.362933 $15$ 0.051696 0.363868
 $T$ Time-Periodic $\overline{s_1'(120)}$ Time-Periodic $\overline{s'_2(120)}$ $0.2$ 0.045093 0.364358 $0.4$ 0.045099 0.364336 $0.6$ 0.045118 0.364294 $0.8$ 0.045158 0.364233 $1.0$ 0.045224 0.364154 $1.1$ 0.056211 0 $1.2$ 0.056282 0 $1.4$ 0.056449 0 $1.5$ 0.056545 0 $1.6$ 0.056648 0 $1.8$ 0.056874 0 $1.9$ 0.045852 0.363676 $2$ 0.045949 0.363617 $5$ 0.048875 0.362510 $10$ 0.050970 0.362933 $15$ 0.051696 0.363868
Average spreading speeds of $u$ and $v$ with $\sigma_1 = \sigma_2 = 1$ and different values of the length of space-period $L$ at time $t = 120$
 Space period $L$ Average speed of $u$ $\overline{s_1'(120)}/\overline{s_1'(300)}/\overline{s_1'(500)}$ Average speed of $v$ $\overline{s'_2(120)}/\overline{s'_2(300)}/\overline{s'_2(500)}$ $0.2$ 0.045131 0.364630 $0.6$ 0.045402 0.366188 $1.0$ 0.045908 0.369344 $1.5$ 0.046770 0.374485 $1.6$ 0.058183 0.013643 ($v$ is vanishing) $1.8$ 0.057517 0.012500 ($v$ is vanishing) $2.0$ 0.058908$^*$ 0.006666$^*$ ($v$ is vanishing) $2.1$ 0.059048$^*$ 0.007000$^*$ ($v$ is vanishing) $2.2$ 0.059164$^*$ 0.007333$^*$ ($v$ is vanishing) $2.3$ 0.047889$^*$ 0.381331$^*$ $2.5$ 0.047982$^*$ 0.382087$^*$ $2.8$ 0.047892$^*$ 0.382278$^*$ $3.0$ 0.047668$^*$ 0.381776$^*$ $3.2$ 0.047319$^*$ 0.380748$^*$ $3.3$ 0.058552$^*$ 0.011000$^*$ ($v$ is vanishing) $3.4$ 0.058332$^*$ 0.011333$^*$ ($v$ is vanishing) $4$ 0.056601$^*$ 0.013333$^*$ ($v$ is vanishing) $5$ 0.052802$^*$ 0.016666$^*$ ($v$ is vanishing) $10$ 0.020768$^{**}$ 0.020000$^{**}$ ($v$ is vanishing) $15$ 0.030000$^{**}$ 0.030000$^{**}$ ($v$ is vanishing) $20$ 0.038608$^{**}$ 0.034420$^{**}$ ($v$ is vanishing) * These average speeds are at t = 300. ** These average speeds are at t = 500.
 Space period $L$ Average speed of $u$ $\overline{s_1'(120)}/\overline{s_1'(300)}/\overline{s_1'(500)}$ Average speed of $v$ $\overline{s'_2(120)}/\overline{s'_2(300)}/\overline{s'_2(500)}$ $0.2$ 0.045131 0.364630 $0.6$ 0.045402 0.366188 $1.0$ 0.045908 0.369344 $1.5$ 0.046770 0.374485 $1.6$ 0.058183 0.013643 ($v$ is vanishing) $1.8$ 0.057517 0.012500 ($v$ is vanishing) $2.0$ 0.058908$^*$ 0.006666$^*$ ($v$ is vanishing) $2.1$ 0.059048$^*$ 0.007000$^*$ ($v$ is vanishing) $2.2$ 0.059164$^*$ 0.007333$^*$ ($v$ is vanishing) $2.3$ 0.047889$^*$ 0.381331$^*$ $2.5$ 0.047982$^*$ 0.382087$^*$ $2.8$ 0.047892$^*$ 0.382278$^*$ $3.0$ 0.047668$^*$ 0.381776$^*$ $3.2$ 0.047319$^*$ 0.380748$^*$ $3.3$ 0.058552$^*$ 0.011000$^*$ ($v$ is vanishing) $3.4$ 0.058332$^*$ 0.011333$^*$ ($v$ is vanishing) $4$ 0.056601$^*$ 0.013333$^*$ ($v$ is vanishing) $5$ 0.052802$^*$ 0.016666$^*$ ($v$ is vanishing) $10$ 0.020768$^{**}$ 0.020000$^{**}$ ($v$ is vanishing) $15$ 0.030000$^{**}$ 0.030000$^{**}$ ($v$ is vanishing) $20$ 0.038608$^{**}$ 0.034420$^{**}$ ($v$ is vanishing) * These average speeds are at t = 300. ** These average speeds are at t = 500.

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