Periodically varying externally imposed environmental effects on population dynamics

Effects of externally imposed periodic changes in the environment on population dynamics are studied with the help of a simple model. The environmental changes are represented by the temporal and spatial dependence of the competition terms in a standard equation of evolution. Possible applications of the analysis are on the one hand to bacteria in Petri dishes and on the other to rodents in the context of the spread of the Hantavirus epidemic. The analysis shows that spatiotemporal structures emerge, with interesting features which depend on the interplay of separately controllable aspects of the externally imposed environmental changes.

Figure 1

The bold curve depicts $u_{\max }$ as the bubble moves adiabatically; acceleration is effectively zero. The remaining curve displays $u_{\max }$ for a bubble accelerating linearly at $0.1a^2{W}_c$.

Figure 2

Temporal evolution of $u_{\max }$ where in (a) $A=8W_c$, (b) $A=10W_c$, and $W=10W_c$ in both. Each curve is characterized by one of seven sinusoidal bubble velocities, $0.1a{W}_c$, $1a{W}_c$, $3a{W}_c$, $5a{W}_c$, $7a{W}_c$, and $10a{W}_c$ where $v_f=a{W}_c$ when $D=a{W}_c^2$, as in the plots above. Higher velocities induce lower maximum population density values and thus drive a population closer to extinction as the amplitude of oscillation increases.

Figure 3

A variety of bowties exhibit the affects for different values of $v$ when a bubble of $A=8W_c$, $W=10W_c$, and $D=a{W}_c^2$ moves with sinusoidal velocity. Velocities from top to bottom range from $a{W}_c\text{to}10a{W}_c$ and increment by $0.5a{W}_c$ where $v_f=a{W}_c$ when $D=a{W}_c^2$. The trajectories are parametric in time such that $u_{\max }$ is increasing at the end points.

Figure 4

The amplitude of the bowtie is plotted against the average velocity (which here varies sinusoidally). The inset shows the average $u_{\max }$ after long time. The combination of the both show the validity of the analytic solution in the high velocity regime when the curves for $D=a{W}_c^2$ and $D=0$ converge.

Figure 5

Amplitude of the front oscillations for several velocities of the bubble where $D=a{W}_c^2$.

Figure 6

Amplitude of the front oscillations for several values of the inverse of the frequency of breathing of the bubble where $D=a{W}_c^2$.

Figure 7

Maximum value adopted by the population as a function of the instantaneous width of the bubble. The velocity of breathing is: (a) $0.1a{W}_c$, (b) $1a{W}_c$, (c) $10a{W}_c$, (d) $40a{W}_c$, and $D=a{W}_c^2$.

Figure 8

After long times a periodic dip in all values of $u_{m}ax$ is observed for constant velocities. The sampling above (A, B, C, D, and E) corresponds to velocities ($3.5aW$, $4.5aW$, $5.5aW$, $6.5aW$, and $8.5aW$, respectively) where $v_f=aW$ because $D=a{W}_c^2$.