On the Geometric Diversity of Wavefronts for the Scalar Kolmogorov Ecological Equation

Abstract

We answer three fundamental questions concerning monostable traveling fronts for the scalar Kolmogorov ecological equation with diffusion and spatiotemporal interaction: These are the questions about their existence, uniqueness and geometric shape. In the particular case of the food-limited model, we give a rigorous proof of the existence of a peculiar, yet substantive and nonlinearly determined class of non-monotone and non-oscillating wavefronts.

Appendix

Theorem 29 follows from Faria and Trofimchuk (2010, Theorems 2.1, 3.8, 4.3; Corollaries 3.9, 3.11). To state this result, we first introduce some notation. For $d=(d_1,\dots ,d_N)\in \mathbb {R}^N$, we say that $d>0$ (respectively, $d\ge 0$) if $d_i>0$ (respectively $d_i\ge 0$) for $i=1,\dots ,N$. In the Banach space $\mathcal{C}=C([-\tau ,0];\mathbb {R}^N)$, we consider the partial orders $\ge ,$ resp. $>,$ defined as follows: $\phi \ge \psi$ if and only if $\phi (\theta )- \psi (\theta )\ge 0$ for $\theta \in [-\tau ,0]$; in an analogous way, $\phi >\psi$ if $\phi (\theta )- \psi (\theta )> 0$ for $\theta \in [-\tau ,0]$. Next, $\mathcal{C}_+$ denotes the positive cone $C([-\tau ,0];[0,\infty )^N)$.

We also will need the following Banach spaces:

$C_0=\{ y\in C_b:\lim _{s\rightarrow \pm \infty }y(s)=0\}$ is considered as a subspace of $C_b$;

$C_\mu =\{ y\in C_b:\sup _{s\le 0}e^{-\mu s}|y(s)|<\infty \}$ (for $\mu >0$) with the norm

$$\begin{aligned} \Vert y\Vert _\mu =\max \{\Vert y\Vert _\infty ,\Vert y\Vert _{\mu }^- \}\quad \mathrm{where} \quad \Vert y\Vert _\mu ^-=\sup _{s\le 0}e^{-\mu s}|y(s)|. \end{aligned}$$

The space $C_{\mu ,0}=C_{\mu }\cap C_0$ will be considered as a subspace of $C_\mu$.

We will analyze singular perturbations of the heteroclinic connection in the system of functional differential equations

$$\begin{aligned} u'(t)=f(u_t),\quad t\in \mathbb {R}, \end{aligned}$$

(52)

where f is such that

(H1):

$f(0)=f(\kappa )=0$, where $\kappa$ is some positive vector;

(H2):

(i) f is $C^2$-smooth; furthermore, (ii) for every $M>0$, there is $\beta >0$ such that $f_i(\varphi )+\beta \varphi _i(0)\ge 0,$ $i=1,\dots ,N$, for all $\varphi \in \mathcal{C}$ with $0\le \varphi \le M$;

(H3):

for Eq. (52), the equilibrium $u=\kappa$ is locally asymptotically stable and globally attractive in the set of solutions of (1.2) with the initial conditions $\varphi \in \mathcal{C}_+,\varphi (0)> 0$;

(H4):

for Eq. (52), its linearized equation about the equilibrium 0 has a real characteristic root $\lambda _0>0$, which is simple and dominant (i.e., $\mathfrak {R}\, z<\lambda _0$ for all other characteristic roots z); moreover, there is a characteristic eigenvector $\mathbf{v}>0$ associated with $\lambda _0$.

Then, the following holds.

Theorem 29

Faria and Trofimchuk (2010) Assuming (H1)–(H4), then Eq. (52) has a positive heteroclinic solution $u^*(t)$: $u^*(-\,\infty )=0, \ u^*(+\,\infty )=\kappa$. Next, denote by $\sigma (A)$ the set of characteristic values for

$$\begin{aligned} u'(t)=Lu_t,\quad \mathrm{where}\quad L=Df(0). \end{aligned}$$

Let positive $\mu <\lambda _0$ be such that the strip $\{ \lambda \in \mathbb {C}: \mathfrak {R}\, \lambda \in (\mu , \lambda _0)\}$ does not intersect $\sigma (A)$. Then, there exists a direct sum representation

$$\begin{aligned} C_{\mu ,0}=X_\mu \oplus Y_\mu , \quad \text{ where } \ X_\mu , Y_\mu \ \text{ are } \text{ subspaces } \text{ of } \ C_{\mu ,0}, \ \ \dim X_\mu =1, \end{aligned}$$

and $\varepsilon ^*>0$, $\sigma >0$, such that for $0<\varepsilon \le \varepsilon ^*$, the following holds: For each unit vector $w\in \mathbb {R}^p$, in a neighbourhood $B_\sigma (0)$ of $u^*(t)$ in $C_\mu$, the set of all wavefronts $u(t,x)=\psi (ct+w\cdot x)$ of

$$\begin{aligned} {{\partial u}\over {\partial t}}(t,x)=\Delta u(t,x)+f(u_t(\cdot ,x)),\quad t\in \mathbb {R},\ x\in \mathbb {R}^p. \end{aligned}$$

with speed $c=1/\varepsilon$ and connecting 0 to $\kappa$ forms a one-dimensional manifold (which does not depend on the choice of $\mu$), with the profiles

$$\begin{aligned} \psi (\varepsilon , \xi ) =u^*+\xi +\phi (\varepsilon , \xi ),\quad \xi \in X_\mu \cap B_\sigma (0), \end{aligned}$$

where $\phi (\varepsilon , \xi ) \in Y_\mu \cap B_\sigma (0)$ is continuous in $(\varepsilon ,\xi )$. Next, the profile $\psi (\varepsilon ,0)$ is positive and satisfies $\psi (\varepsilon ,0)\rightarrow u^*\ \mathrm{in} \ C_\mu$ as $\varepsilon \rightarrow 0^+$. Moreover, the components of the profile $\psi (\varepsilon , 0)$ are increasing in the vicinity of $-\,\infty$ and $\psi (\varepsilon ,0)(t)=O(e^{\lambda (\varepsilon )t}),$ $\psi '(\varepsilon ,0)(t)=O(e^{\lambda (\varepsilon )t})$ at $-\,\infty$, where $\varepsilon =1/c$ and $\lambda (\varepsilon )$ is the real solution of

$$\begin{aligned} \det \Delta _\varepsilon (z)=0,\quad \mathrm{}\quad \Delta _\varepsilon (z):=\varepsilon ^2 z^2I-zI+L(e^{z\cdot }I), \end{aligned}$$

where $L=Df(0)$, with $\lambda (\varepsilon )\rightarrow \lambda _0$ as $\varepsilon \rightarrow 0^+$.

Remark 30

In fact, after a slight modification of the proof of Theorem 3.8 in Faria and Trofimchuk (2010), one can note that the conclusions of Theorem 29 remain valid if we replace the space $C_{\mu ,0}$ with $C_{\mu ,\delta }=\{ y\in C_b:\Vert y\Vert _{\mu ,\delta } <\infty \}$ for small $\delta \in (0,\mu )$ and with the norm

$$\begin{aligned} \Vert y\Vert _{\mu ,\delta } =\max \{\Vert y\Vert _\delta ^+,\Vert y\Vert _{\mu }^- \}\quad \mathrm{where} \quad \Vert y\Vert _\mu ^-=\sup _{s\le 0}e^{-\mu s}|y(s)|, \ \Vert y\Vert _\delta ^+=\sup _{s\ge 0}e^{\delta s}|y(s)|. \end{aligned}$$

To see this, it suffices to use the change of variables $\phi (t) = u^*(t)+ e^{-\delta t}w(t)$ instead of $\phi (t) = u^*(t)+ w(t)$ before formula (3.3) in Faria and Trofimchuk (2010). As a consequence of this observation, there exist small $\epsilon _0 >0$ and some constant $C>0$ which does not depend on $\epsilon$ such that $|\psi (\varepsilon ,0)(t) -\kappa | \le Ce^{-\delta t}$, $t\ge 0$, $\epsilon \in [0,\epsilon _0]$.