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# Dynamical stabilization and traveling waves in integrodifference equations

VL and FL are supported by respective Discovery Grants from the Natural Sciences and Engineering Research Council (NSERC) of Canada (RGPIN-2016-04318 and RGPIN-2016-04795). FL is grateful for a Discovery Accelerator Supplement award from NSERC (RGPAS 492878-2016). FL thanks the participants of the workshop "Integrodifference equations in spatial ecology: 30 years and counting" (16w5121) at the Banff International Research Station for their feedback on an oral presentation of this material

• Integrodifference equations are discrete-time analogues of reaction-diffusion equations and can be used to model the spatial spread and invasion of non-native species. They support solutions in the form of traveling waves, and the speed of these waves gives important insights about the speed of biological invasions. Typically, a traveling wave leaves in its wake a stable state of the system. Dynamical stabilization is the phenomenon that an unstable state arises in the wake of such a wave and appears stable for potentially long periods of time, before it is replaced with a stable state via another transition wave. While dynamical stabilization has been studied in systems of reaction-diffusion equations, we here present the first such study for integrodifference equations. We use linear stability analysis of traveling-wave profiles to determine necessary conditions for the emergence of dynamical stabilization and relate it to the theory of stacked fronts. We find that the phenomenon is the norm rather than the exception when the non-spatial dynamics exhibit a stable two-cycle.

Mathematics Subject Classification: Primary: 37L15, 92D25; Secondary: 37N25.

 Citation: • • Figure 2.  Solution of the integrodifference equation (left) and its 'phase plane' (right), where $F$ is the Ricker function with $r = 0.8$ (solid) and $r = 1.03$ (dashed) and $K$ is the Laplace kernel with $a = 15$

Figure 1.  Numerical solution of the IDE in 1, plotted for even (top panel) and odd (bottom panel) generations every 10 time steps. The solid lines correspond to the Ricker growth function 2 with $r = 2.2$ and the dashed line to the logistic growth function 3 with $r = 2.44.$ The dispersal kernel is the Laplace kernel 4 with $a = 15.$ The initial condition was the function $N_0(x) = n_-\chi_{x\leq 10}.$

Figure 3.  Plot of the implicit functions defined by equations 15 (thin blue lines) and 16 (thicker red line) with $c = c^*$ for the Ricker function with $r = 1.0327$ (left) and $r = 2.526$ (right). Note that there are no negative real roots as we chose $r>r^*.$ Only the upper half plane is plotted; the lower half plane is symmetric

Figure 4.  Solution of the integrodifference equation (left) and its phase plane (right), where $F$ is the Ricker function with $r = 1.8$ and $K$ is the Laplace kernel with $a = 15$

Figure 5.  Phase plane of the solution in Figure 1. The solid curve corresponds to the Ricker function, the dashed curve to the logistic function, both with parameter $r = 2.2.$ $K$ is the Laplace kernel with $a = 15.$

Figure 6.  Solution of the integrodifference equation (left) and its phase plane (right), where $F$ is the Ricker function with $r = 2.525$ and $K$ is the Laplace kernel with $a = 15$

Figure 7.  Solution of the integrodifference equation (left) and its phase plane (right), where $F$ is the logistic function with $r = 2.5$ and $K$ is the Laplace kernel with $a = 15$

Figure 8.  Solution of the IDE, with Ricker function and Laplace kernel ($a = 15$). The growth parameter is $r = 2.6,$ so that the two-cycle $n_\pm$ is unstable for the Ricker dynamics and a stable four-cycle exist (denoted by $n^-_-, n^-_+, n^+_-, n^+_+$). Initial conditions are $N_0 = n^+_+\chi_{[x\geq 10]}.$

Table 1.  Dynamic behavior of the map $N \mapsto F(N)$ with $F$ as in 2 or 3. The abbreviation 'g.a.s.' stands for globally asymptotically stable within all non-stationary, non-negative solutions

 Dynamic behavior Ricker function 2 Logistic function 3 $N^*=1$ g.a.s. $0< r< 1$ $0< r< 1$ monotone approach $N^*=1$ g.a.s. $1< r< 2$ $1< r< 2$ oscillatory approach $N^*=1$ unstable $2< r< 2.526$ $2 Table 2. Shape of the traveling profile emerging from$ N^* = 0 $in IDE 1 as a function of parameter$ r $for the Ricker and the logistic function and with Laplace dispersal kernel. When the kernel has compact support, monotone traveling waves may not exist even if they do with a Laplace kernel   Shape of the traveling profile Ricker function Logistic function Monotone on$ [0,1]  0< r< 1.0327  0< r< 1.0686 $Damped oscillations at$ N=1  1.0327< r< 2.5072  1.0686< r< 2.570 $Wavetrain around$ N = 1  2.5072< r< 2.692 \$ NA
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