Periodic Traveling Waves in an Integro-Difference Equation With Non-Monotonic Growth and Strong Allee Effect

For $i=1,2$, define the right inverse of $w_{i}$ by

Note that $\Phi_{i}(\ell)$ is finite for $0<\ell<||w_{i}||_{\infty}$, where $||w_{1}||_{\infty}=1$, and $a<||w_{2}||_{\infty}<b$ by the assumption of this theorem.

By the translation invariance property of $Q$, it suffices to show

and

for all $x\in\mathbb{R}$. The first equation follows immediately from definition (2.2). To prove the second, observe that $w_{1}$ is monotonically decreasing and satisfies $w_{1}(\infty)=0$ and $w_{1}(-\infty)=1$ (see for example Figure 2). Thus,

Applying the integro-difference operator yields

We can then differentiate with respect to $x$:

From (H4), $\frac{\,\mathrm{d}w_{2}}{\,\mathrm{d}x}$ has at most 1 sign change; thus, $w_{2}$ has at most 1 turning point. We also have $w_{2}(x)<b$ for all $x$, hence $g(w_{2}(x))\in\{0,1\}$.

We claim

where $c^{*}$ is defined by (2.3). This can be shown by arguing in cases depending on the number of turning points of $w_{2}$.

1. Case 1.

   If $w_{2}$ has no turning points, then $w_{2}$ is monotone decreasing with $0\leq w_{2}(x)\leq m$ for all $x$. It follows by the intermediate value theorem (IVT) that $c^{*}=\Phi_{2}(a)$ satisfies (2.8).

2. Case 2.

   If $w_{2}$ has one turning point, say $x_{0}\in\mathbb{R}$, then $w_{2}$ must be increasing for $x<x_{0}$ and decreasing for $x>x_{0}$. It follows that $x_{0}$ is a global maximum, and $w_{2}$ is increasing on $(-\infty,x_{0})$ and decreasing on $(x_{0},\infty)$. We have $g(w_{2}(x))=1$ for $x<x_{0}$. By the IVT, $c^{*}=\Phi_{2}(a)$ is the unique solution to (2.8).

Taking the convolution of equation (2.8) with $k(x)$ yields

This completes the proof. ∎

By Theorem 2.2, we have

for every $\ell\in(0,m)$. Taking the first difference yields

∎

We define two continuous functions $\tilde{w_{1}},\tilde{w_{2}}:\mathbb{R}\times[0,\infty)\to\mathbb{R}$ by

and

Observe that $\tilde{w_{1}}$ is symmetric with respect to $x=-r$, and $\tilde{w_{2}}$ is symmetric with respect to $x=-r-c^{*}$. This notation can be justified by observing that $\tilde{w_{1}}(x,r)\to w_{1}(x)$ and $\tilde{w_{2}}(x,r)\to w_{2}(x)$ as $r\to\infty$ for all $x$. We also set

$\phi_{1}$ and $\phi_{2}$ are the right-inverse of $\tilde{w_{1}}$ and $\tilde{w_{2}}$, respectively. For a fixed $r>0$, the value of $\phi_{i}(r,\ell)$ is finite if $0<\ell<||\tilde{w}_{i}(\,\cdot\,,r)||_{\infty}$, with the latter bound converging to $||w_{i}||_{\infty}$ as $r\to\infty$. They satisfy

We now apply the growth function to $\tilde{w_{1}}$, assuming $r$ is sufficiently large enough so that $a<||\tilde{w_{1}}(\,\cdot\,,r)||_{\infty}<b$; this is guaranteed by taking $r>\Phi_{1}\left(\frac{1-a}{2}\right)$. We have

Note that the above expression converges to $g(w_{1}(x))$ as $r\to\infty$ for all $x$. We then apply the convolution operator:

Computing the derivative of this expression yields

By (H5), the number of turning points of $\tilde{w_{2}}$ is at most 3 (assuming $r$ is sufficiently large). Furthermore, $\tilde{w_{2}}$ is symmetric around $x=-r-c^{*}$; thus the sign changes must come in pairs or occur exactly at $x=-r-c^{*}$. We can immediately exclude the cases of 0 and 2 turning points because $\tilde{w_{2}}$ is non-negative, vanishes at $\pm\infty$, and is not identically zero. As in the proof of Theorem 2.2, this leaves two cases:

1. Case 1.

   If $\tilde{w_{2}}$ has 1 turning point, then it must occur at $x=-r-c^{*}$. We have $\tilde{w_{2}}(-r-c^{*},r)=w_{2}(-r-c^{*})-w_{2}(r+c^{*})\to m$ as $r\to\infty$; hence, it is sufficient to take $r$ large enough so that $|\tilde{w_{2}}(-r-c^{*},r)-m|<\min\{|a-m|,|b-m|\}$. It follows by the triangle inequality that $\tilde{w_{2}}(x,r)\leq\tilde{w_{2}}(-r-c^{*},r)<b$ for all $x$.

2. Case 2.

   If $\tilde{w_{2}}$ has 3 turning points, then again by symmetry, they must be given by $\{-r-c^{*}-t,-r-c^{*},-r-c^{*}+t\}$, for some $t>0$. We can label them in increasing order by $t_{1}<t_{2}<t_{3}$. It follows that $t_{1}$ and $t_{3}$ are maxima with $\tilde{w_{2}}(t_{1},r)=\tilde{w_{2}}(t_{3},r)\to||w_{2}||_{\infty}$ as $r\to\infty$, and $t_{2}$ is a local minima with $\tilde{w_{2}}(t_{2},r)\to m$ as $r\to\infty$. Thus $\tilde{w_{2}}$ is monotone increasing on $(-\infty,t_{1})\cup(t_{2},t_{3})$ and monotone decreasing on $(t_{1},t_{2})\cup(t_{3},\infty)$. By taking $r$ sufficiently large, we can make $\left|\tilde{w_{2}}(t_{1},r)-||w_{2}||_{\infty}\right|=\left|\tilde{w_{2}}(t_{3},r)-||w_{2}||_{\infty}\right|<\left|b-||w_{2}||_{\infty}\right|$ and $\left|\tilde{w_{2}}(t_{2},r)-m\right|<\left|a-m\right|$.

In both cases, we have $a\leq\tilde{w_{2}}(x,r)\leq b$ on a closed interval of radius $r+c^{*}+\phi_{2}(r,a)$ and $\tilde{w_{2}}(x,r)<a$ elsewhere. Taking composition with $g$ yields

Hence,

Shifting the expression left by $c^{*}+\phi_{2}(r,a)$ units yields

We are now prepared to prove the theorem using an inductive argument. Let $A$ be the level set defined in the statement of the theorem. Without loss of generality, we write $A=[-r_{0},r_{0}]$, for some $r_{0}>0$. We have

for all $x$. Applying the convolution operator yields

Let $(r_{n})_{n=0}^{\infty}$ be a sequence of real numbers satisfying

If $r_{0}$ is sufficiently large, we can show that $r_{n}$ is strictly increasing and unbounded. This is due to our assumption that $c^{*}>0$. Since $\phi_{2}(r,a)\to c^{*}$, it suffices to assume $|\phi_{2}(r,a)-c^{*}|<c^{*}$, i.e. $r_{0}$ is sufficiently large so that $\phi_{2}(r,a)>0$ for all $r>r_{0}$. Plugging this into the recurrence (2.18) yields $r_{n+1}-r_{n}=c^{*}+\phi_{2}(r,a)>c^{*}>0$.

This argument shows that, assuming $r_{0}$ is sufficiently large, equation (2.17) may be applied recursively. From the principle of induction, it follows that $u_{2n+1}(x)=\tilde{w_{1}}(x-r_{n},r_{n})$ for all $n\geq 0$. Knowing that $(r_{n})$ is unbounded allows us to use equation (2.18) to compute the asymptotic spreading speed as the following limit.

This completes the proof. ∎