Traveling wave fronts in a single species model with cannibalism and strongly nonlocal effect

1.   Introduction

In this paper, we are concerned with the following nonlocal reaction diffusion equation

for $-\infty < x < +\infty$ and $t > 0$, where the parameter $r > 0, K > 0, h > 0$ and the spatiotemporal convolution $f*u$ is defined by

and the kernel $f$ satisfies the usual normalization assumption, namely

so that the kernel does not affect the spatially uniform steady states, which in this model will be the extinction state $u\equiv 0$ and the positive equilibrium $u\equiv u^* = \frac{Kr}{r+Kh}$. This type of equation was introduced in ^[1] to model the dynamical behavior of a single species. The parameters $r, K, h$ represent the intrinsic growth rate of the species, the capacity of the environment, and cannibalism rate, respectively. More specifically, the term $hu^2$ signifies intraspecific cannibalism, which is a widespread phenomenon in a variety of animals. Not only could it result in an increase in death rate, but it could also possess the potential of regulating population size. The convolution term $\frac{(f*u)(x, t)}{K}$ signifies the nonlocal consumption of the resources. For the specific biological background to the model, please refer to ^[2,3].

When the kernel $f$ is taken to be $f(x, t) = \frac{1}{\sqrt{4\pi \rho}}e^{-\frac{x^2}{4\rho}}\delta(t)$, $\rho$ represents the nonlocal effect. By applying super-sub solution method ^[4,5,6] as well as Leray-Schauder topological degree theory ^[7], Zhang and Li ^[1] proved that there exist traveling wave fronts connecting the equilibrium $u\equiv 0$ to the positive equilibrium $u\equiv u^* = \frac{Kr}{r+Kh}$ when the wave speed $c\geq 2\sqrt{r}$. In the present paper, we are interested in traveling front solutions for another particular class of kernels of the form

in which the parameter $\rho$ is representative of the nonlocal effect. For such a type of kernel function (1.3), the nonlinear convolution term $\frac{(f*u)(x, t)}{K}$ implies that the individuals in the population consume the resources not only at the point where they are located but also in some area around this point. We shall consider the traveling wave problem of Eq (1.1) when $\frac{1}{\rho}\ll 1$, i.e., the parameter $\rho$ is sufficiently large, which signifies the strongly nonlocal effect. As pointed out in ^[6], this can be understood as the limit of a highly mobile resource in which the population is represented by $u$ feeds. Note that this particular choice of kernel like (1.3) is the Green's function for an ordinary differential equation, we can rewrite Eq (1.1) as the coupled reaction-diffusion equations. Thus, the phase space of the system for the traveling wave problem corresponding to Eq (1.1) is four-dimensional. A traveling wave front can be characterized as a heteroclinic connection in this phase space, and then the dynamical systems theory, especially the geometric singular perturbation theory and Fenichel's invariant manifold theory ^[8,9,10], can be successfully used to establish the existence of such a connection.

The remaining part of this paper is organized as follows. In Section 2, we formulate the traveling wave problem of system (1.1) with the kernel (1.3) from the viewpoint of the dynamical system, which can be viewed as a singular perturbation problem when $\rho$ is taken to be a sufficiently large perturbed parameter. In Section 3, by analyzing the dynamics of limiting slow and limiting fast systems for the singular perturbation problem, we give a singular heteroclinic orbit in the phase space of the traveling wave system of Eq (2.4), which is composed of the solutions of limiting slow and fast systems. In Section 4, we employ geometric singular perturbation theory and Fenichel's invariant manifold theory to show that the above singular heteroclinic orbit persists if the parameter $\rho$ is taken to be sufficiently large. Finally, we summarize our results in Section 5.

2.   Geometric singular perturbation formulation for traveling wave problem

In this section, we will formulate the traveling wave problem of system (1.1) as a geometric singular perturbation problem.

First, if we define $w = f*u$, namely,

it is straightforward to see that $w$ satisfies

and, thus the integrodifferential equation (1.1) can be rewritten as the following coupled reaction-diffusion system

Let $\varepsilon = \frac{1}{\rho}$, then $\varepsilon$ is sufficiently small if $\rho$ is sufficiently large. Our objective now is to establish the existence of traveling wavefront solutions of (2.3) connecting the two uniform steady-states $(u, w) = (0, 0)$ and $(u^*, u^*)$, for sufficiently small $\varepsilon$. Converting to traveling wave form, by setting

we have

where

Note that system (2.4) is invariant under the transformation $(c, z)\mapsto (-c; -z)$ and thus we may assume, without loss of generality, that $c > 0$. Upon introducing the two new variables $V: = \frac{dU}{dz}$ and $Y: = \varepsilon^{-1}\frac{dW}{dz}$, system (2.4) can be reformulated as

which is called the fast system provided that $\varepsilon$ is sufficiently small. In terms of the slow scale $\xi: = \varepsilon z$, the corresponding slow system of (2.6) becomes

Thus, traveling wave fronts of (1.1) correspond to heteroclinic orbits of the fast system (2.6) or the slow system (2.7) connecting its two equilibrium points, that is,

3.   Properties of limiting systems

In this section, we consider the fast and slow systems (2.6) and (2.7) from the geometric singular perturbation point of view. When $\varepsilon = 0$, we have the following limiting fast and limiting slow systems

and

Thus, the critical manifold $S$ is given by

which is the set of equilibria of the limiting fast system (3.1). This critical manifold $S$ consists of the two two-dimensional manifolds $S_1, S_2$, which can be parameterized by the slow variables $W$ and given by

and

Moreover, the manifolds $S_1$ and $S_2$ intersect along the line $W = \frac{1}{b}$. See Figure 1 for a schematic depiction of the two manifolds $S_1, S_2$ and the heteroclinic orbit associated to the traveling wave front.

3.1.   limiting slow system

Next, we study the reduced dynamics on the critical manifold $S$. It follows from the limiting slow system (3.2) that the reduced dynamics on the manifold $S_1$ is determined by the linear system

since $U = 0$ on $S_1$. The solutions of system (3.6) can be directly solved by

for arbitrary constants $C_1, C_2$. Similarly, the reduced dynamics on the manifold $S_2$ are determined by the linear system

since $U = \frac{1}{a}(1-bW)$ on $S_2$. The solutions of system (3.8) can be directly solved by

for arbitrary constants $C_3, C_4$. These constants $C_i(i = 1, 2, 3, 4)$ are determined by the asymptotic boundary conditions (2.8) and by the dynamics of layer problem (3.1). Following the ideas used in ^[11], we divide our spatial domain into three fields (with respect to the slow variable $\xi$): two slow fields $I_s^-, I_s^+$ which are away from the layer dynamics and one fast field $I_f$ which is near the layer dynamics. Without loss of generality, we assume here that the layer dynamics are centered around zero. Thus, these fast and slow fields can be chosen as follows

where $I_f$ corresponds to the layer dynamics from $S_2$ to $S_1$, while $I_s^-$ and $I_s^+$ correspond to the reduced dynamics on $S_2$ and $S_1$, respectively. With the asymptotic scaling $\varepsilon^{\frac{1}{2}}$, we choose to ensure that it is asymptotically small with respect to the slow variable $\xi$ and asymptotically large with respect to the fast variable $z: = \varepsilon^{-1}\xi$. In fact, it is not hard to find that $\varepsilon^{\frac{1}{2}}\ll 1$ and $\varepsilon^{\frac{1}{2}-1}\gg 1$.

According to the asymptotic boundary conditions (2.8), the heteroclinic orbit associated to the traveling front solution should approach $A^-$ as $\xi\rightarrow -\infty$. So, the critical manifold of interest is $S_2$ for $\xi\in I_s^-$ (see the top frame of Figure 2). Thus, the slow variable $W$ and $Y$ are determined by (3.9). Note that $W(-\infty) = u^*, Y(-\infty) = 0$, then we can derive that $C_4 = 0$. Similarly, for $\xi\in I_s^+$, the critical manifold of interest is $S_1$ (see the bottom frame of Figure 2), and the slow variables $W$ and $Y$ are determined by (3.7). The boundary condition $W(+\infty) = Y(+\infty) = 0$ yields that $C_1 = 0$. Consequently, the solutions (3.7) and (3.9) become

and

During the transition through the fast field $I_f$, the evolution equations for the slow variables $W$ and $Y$ are given by

Note that $\varepsilon\ll \varepsilon^{\frac{1}{2}}$, hence the changes of both $W$ and $Y$ are, to leading order, constant during the transition through the fast field $I_f$. In other words, both $W$ and $Y$ should match to leading order at zero, i.e., $W(0^-) = W(0^+), Y(0^-) = Y(0^+)$. By substituting this into (3.11) and (3.12), the two remaining constants $C_2$ and $C_3$ can be determined and given by

Therefore, we have

and

where

It is easily seen that the fast transition always occurs at $W = W_0$. Furthermore, by combining (3.4)–(3.5) with (3.14)–(3.15), the leading order profiles for other variables in the slow fields can now be successfully obtained. In particular, we have

and

3.2.   Limiting fast system

Now, we turn to study the layer dynamics in the fast field $I_f$. In fact, the dynamics of the heteroclinic orbit are, to leading order, determined by the limiting fast system (3.1), and the orbit has to transit from $S_2$ to $S_1$. Note that $W$ is to leading order constant in the fast field, i.e., $W = W_0$ in $I_f$. Consequently, the two-components $(U, V)$ equations in the limiting fast system with $W = W_0$ becomes

where

Obviously, system (3.17) can be rewritten as the following two-order ODE

which is exactly the traveling wave equation for the classical Fisher-KPP equation. It is well-known that system (3.17) admits a heteroclinic orbit connecting its two uniform steady-state $(U, V) = (0, 0)$ and $(U, V) = (\frac{L}{a}, 0)$. Moreover, system (3.19) has a unique monotonic decreasing solution if and only if,

where $c_m$ is the so-called minimum wave speed for the associated Fisher-KPP equation. This restriction on $c$ arises from the fact that the equilibrium point $(U, V) = (0, 0)$ in the planar system (3.17) will change from a stable node to a stable focus as $c$ decreases past $c_m$, so that $U$ becomes negative for sufficiently large $z$ when $c < c_m$. In particular, for sufficiently large $z\gg 1$, we have

where

For more details, please refer to ^[12] and references therein.

3.3.   Singular heteroclinic orbit in the singular limit $\varepsilon\rightarrow 0$

Based on the above analyses on limiting slow and limiting fast systems, we are now able to construct singular heteroclinic orbit in the singular limit $\varepsilon\rightarrow 0$.

Let's denote by $U(A^-)$ (respectively, $S(A^+)$) the unstable (respectively, stable) manifold of $A^- = (u^*, 0, u^*, 0)$ (respectively, $A^+ = (0, 0, 0, 0)$) on $S_2$ (respectively, $S_1$). It follows from (3.14)–(3.15) that $U(A^-)$ and $S(A^+)$ can be explicitly represented as

and

It is easy to see that both $U(A^-)$ and $S(A^+)$ are straight lines. Let $\Lambda_-$ be the limiting slow orbit from $A^-$ to $B^-: = (\frac{L}{a}, 0, W_0, -W_0)\in S_2$, and $\Lambda_+$ be the limiting slow orbit from $B^+: = (0, 0, W_0, -W_0)\in S_1$ to $A^+$. Then, we have

and

Let $\Gamma$ be the limiting fast orbit from $B^-$ to $B^+$, which is determined by system (3.17). In fact, we can find that $\Gamma$ is a curve locating at the two-dimensional plane $\pi: = \left\{(U, V, W, Y)\in \mathbb{R}^4\left|\, W = W_0, Y = -W_0\right.\right\}$. Thus, the curve segment

is the singular heteroclinic orbit from $A^-$ to $A^+$ in the singular limit $\varepsilon\rightarrow 0$. See also Figure 1 for a schematic depiction.

4.   Persistence of singular heteroclinic orbit for $0 < \varepsilon \ll 1$

In this section, we show the persistence of singular heteroclinic orbit for sufficiently $0 < \varepsilon \ll 1$ in system (2.6) or (2.7) and thus the existence of traveling wave fronts in Eq (1.1). We summarize the main results of this paper as follows.

Theorem 4.1. For any fixed $c\geq 2\sqrt{rL}$, where $L = \sqrt{\frac{Kh}{r+Kh}}$, and for the case when the kernel $f$ is given by (1.3), Eq (1.1) possesses a traveling front solution $u(x, t) = U(x-ct)$ satisfying $U(-\infty) = u^*$ and $U(+\infty) = 0$, provided that the nonlocal parameter $\rho$ is sufficiently large.

Proof. Notice that $W$ is given by (3.14), then we have that $W\neq \frac{1}{b}$ along the singular heteroclinic orbit. Thus, both the manifold $S_1$ and $S_2$ are normally hyperbolic along the singular orbit and this singular orbit is a heteroclinic connection between $S_2$ and $S_1$. It follows from Fenichel's invariant manifold theory ^[9] that, for $\varepsilon$ sufficiently small and after appropriately compactifying $S_1$ and $S_2$, there exist locally invariant slow manifold $S_{1, \varepsilon}$ and $S_{2, \varepsilon}$ in the system (2.6) or (2.7) that are $\mathcal{O}(\varepsilon)-$ close to $S_1$ and $S_2$, respectively. Moreover, system (2.6) or (2.7) also admits locally invariant stable and unstable manifolds $\mathcal{W}^s(S_{1, \varepsilon})$ and $\mathcal{W}^u(S_{2, \varepsilon})$ which are $\mathcal{O}(\varepsilon)-$ close to $\mathcal{W}^s(S_1)$ and $\mathcal{W}^u(S_2)$, respectively. Notice that $\Lambda_-\in S_2, \Lambda_+\in S_1$, then $\mathcal{W}^s(\Lambda_+)$ and $\mathcal{W}^u(\Lambda_-)$ possess the similar properties as $\mathcal{W}^s(S_1)$ and $\mathcal{W}^u(S_2)$, respectively. Note that the singular orbit $\Lambda: = \Lambda_- \cup\Gamma\cup\Lambda_+$ is contained in the intersection $\mathcal{W}^s(\Lambda_+) \bigcap\mathcal{W}^u(\Lambda_-)$, and it follows that this singular orbit will persist for sufficiently small $0 < \varepsilon \ll 1$ if the intersection $\mathcal{W}^s(\Lambda_+) \bigcap\mathcal{W}^u(\Lambda_-)$ is transversal along the limiting fast orbit $\Gamma$. In fact, first we can derive from the signs of eigenvalues presented in Subsection 3.1 that $dim(\mathcal{W}^s(\Lambda_+)) = 2+1 = 3$ and $dim(\mathcal{W}^u(\Lambda_-)) = 1+1 = 2$, then it implies that they might intersect along a one-dimensional curve in four-dimensional phase space $\mathbb{R}^4$. Moreover, we can observe that the tangent space $T\mathcal{W}^s(\Lambda_+)$ along $\Gamma$ is given by

where the two vectors $(1, \lambda_1^1, 0, 0)^T, (1, \lambda_2^1, 0, 0)^T$ are composed of the two stable eigenvectors respectively, appended with two $0$ components representing $W, Y$ components which remain constants through the fast transition; while the latter one vector $(0, 0, 1, -1)^T$ represents the direction of $\Lambda_+$. Also, we can observe that the tangent space $T\mathcal{W}^u(\Lambda_-)$ along $\Gamma$ is given by

where the vector $(1, \lambda_1^2, 0, 0)^T$ is composed of the unstable eigenvectors appended with two $0$ components representing $W, Y$ components which remain constants through the fast transition; while the latter one vector $(\frac{L}{a}-u^*, 0, W_0-u^*, -W_0)^T$ represents the direction of $\Lambda_-$. One can easily verify that the vector $(\frac{L}{a}-u^*, 0, W_0-u^*, -W_0)^T$ is linearly independent to the three vectors that span $T\mathcal{W}^s(\Lambda_+)$. Hence, at any points along the limiting fast orbit $\Gamma$, the combined tangent spaces $T\mathcal{W}^s(\Lambda_+)$ and $T\mathcal{W}^u(\Lambda_-)$ contain the full tangent space $T(\mathbb{R}^4)$ to the phase space $\mathbb{R}^4$. Thus, it shows the transversality of the intersection $\mathcal{W}^s(\Lambda_+) \bigcap\mathcal{W}^u(\Lambda_-)$, which can ensure the persistence of heteroclinic connection for $0 < \varepsilon \ll 1$. More precisely, for $0 < \varepsilon \ll 1$, $\Lambda_-, \Lambda_+, \Gamma$ persist. Denote by $\Lambda_-^{\varepsilon}, \Lambda_+^{\varepsilon}, \Gamma^{\varepsilon}$ the perturbed objects, respectively. Thus, the orbit $\Lambda^{\varepsilon}: = \Lambda_-^{\varepsilon} \cup\Gamma^{\varepsilon}\cup\Lambda_+^{\varepsilon}$ corresponds to the singular orbit connecting $A^- = (u^*, 0, u^*, 0)$ to $A^+ = (0, 0, 0, 0)$ with $\Lambda^{\varepsilon}\rightarrow \Lambda$ as $\varepsilon\rightarrow 0$. In view of (2.5), (3.16), (3.18) and (3.20), we can see that, for given parameter conditions presented in Theorem 4.1, Eq (1.1) possesses a traveling front solution $u(x, t) = U(x-ct)$ connecting its extinction state $u = 0$ with the positive equilibrium $u = u^*$ when $\rho$ is sufficiently large. The proof is completed. □

5.   Conclusions and discussion

In this work we deal with the traveling wave problem for a single species model with cannibalism and nonlocal effect. By employing geometric singular perturbation theory and Fenichel's invariant manifold theory, we have proved that, for the case of strongly nonlocal effect, this model admits a traveling front solution going from the extinction state to the positive equilibrium state. It should be remarked here that, for the case of weak nonlocal effect (i.e., $\rho$ is sufficiently small), the traveling wave problem for Eq (1.1) can also be reformulated as a singular perturbation problem. In fact, our fast and slow systems (2.6) and (2.7) can be rewritten as

and

respectively. It is easily seen that if the parameter $\rho$ is taken to be sufficiently small, systems (5.1) and (5.2) become the singular perturbed slow and fast system, respectively. Following the ideas in ^[13,14], in which the traveling waves for the similar models as Eq (1.1) were studied, we can also establish the existence of traveling wave front of Eq (1.1) connecting its two uniform steady states for sufficiently small $\rho$. However, for the model (1.1), the wave speed of the traveling wave front has to satisfy the condition $c\geq 2\sqrt{r}$ if $\rho$ is taken to be sufficiently small, in contrast to the case that $\rho$ is taken to be sufficiently large, in which the wave speed satisfies the condition $c\geq 2\sqrt{rL}$. Note that $L < 1$, i.e., $2\sqrt{rL} < 2\sqrt{r}$, so we believe that Eq (1.1) admits traveling wave front connecting its two uniform steady states for all $\rho > 0$, provided that the wave speed satisfies the condition $c\geq 2\sqrt{r}$.

Furthermore, the methodology of embedding the traveling wave problem into a slow-fast structure and subsequently studying the corresponding dynamics of the limiting slow and limiting fast systems can also be extended to study the higher dimensional traveling wave problems. For instance, one can extend this method to investigate the existence problem of heteroclinic traveling wave connecting two stable rest states for the following singularly perturbed reaction-diffusion equations modeling the evolution of three competing species

where the constants $r_2, r_3, d$ are strictly positive and $0 < \epsilon\ll 1$ (i.e., the species $u_2$ and $u_3$ diffuse very slowly relative to $u_1$), while the two rest states are defined by

We should mention that the existence of a non-monotone traveling wave connecting the rest states $P_2$ and $P_3$ for (5.3) has been established by applying the Conley index theory in ^[15]. However, it is believed that the same results can be demonstrated rigorously by using geometric singular perturbation theory and Fenichel's invariant manifold theory.

Finally, in this work we only established the existence of the traveling wave front solution for Eq (1.1), and the stability of this traveling wave front solution is not considered. A natural question arises regarding how to study the stability properties of this traveling wave front. We think that a potential approach is to combine the singular limit eigenvalue problem (SLEP) method used in ^[16,17,18] with the Evans function method developed in ^[19,20,21] to compute eigenvalues. We leave these extensions for future analysis.

Author contributions

Xijun Deng: Writing-original draft preparation, methodology; Aiyong Chen: Writing-review and editing. All authors have read and agreed to the published version of the manuscript.

Acknowledgments

The first author was partially supported by the Scientific Research Fund of Hunan Provincial Education Department (No.21A0414). The second author was supported by National Natural Science Foundation of China (No.11671107).

Conflict of interest

The authors declare no conflicts of interest regarding the publication of this paper.