The Fisher-KPP nonlocal diffusion equation with free boundary and radial symmetry in $  {\mathbb R}^3 $

Yihong Du; Wenjie Ni; Yihong Du; Wenjie Ni

doi:10.3934/mine.2023041

Mathematics in Engineering

2023, Volume 5, Issue 2: 1-26. doi: 10.3934/mine.2023041

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The Fisher-KPP nonlocal diffusion equation with free boundary and radial symmetry in ${\mathbb R}^3$

Yihong Du ^,,
Wenjie Ni

School of Science and Technology, University of New England, Armidale, NSW 2351, Australia

Received: 13 March 2022 Revised: 06 June 2022 Accepted: 06 June 2022 Published: 16 June 2022

This paper is concerned with the radially symmetric Fisher-KPP nonlocal diffusion equation with free boundary in dimension 3. For arbitrary dimension $N\geq 2$ , in ^[18], we have shown that its long-time dynamics is characterised by a spreading-vanishing dichotomy; moreover, we have found a threshold condition on the kernel function that governs the onset of accelerated spreading, and determined the spreading speed when it is finite. In a more recent work ^[19], we have obtained sharp estimates of the spreading rate when the kernel function $J(|x|)$ behaves like $|x|^{-\beta}$ as $|x|\to\infty$ in ${\mathbb R}^N$ ( $N\geq 2$ ). In this paper, we obtain more accurate estimates for the spreading rate when $N = 3$ , which employs the fact that the formulas relating the involved kernel functions in the proofs of ^[19] become particularly simple in dimension $3$ .

Keywords:

Citation: Yihong Du, Wenjie Ni. The Fisher-KPP nonlocal diffusion equation with free boundary and radial symmetry in ${\mathbb R}^3$ [J]. Mathematics in Engineering, 2023, 5(2): 1-26. doi: 10.3934/mine.2023041

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Abstract

1. Introduction

In a tritrophic food chain, successive predation occurs between three species in the order prey-mesopredator-superpredator, in which there is a transfer of energy and nutrients when one organism eats another. The importance of the coexistence of species in the food chain, lies in avoiding overpopulation and extinction of species. There are several interesting results on the study of these systems involving Holling-type functional responses, and among them we mention: Rao and Narayan ^[1] studied the stability of the interior equilibrium point using the Routh-Hurwitz criterion and the global stability of a three-species food chain model with harvesting by constructing an appropriate Lyapunov function. Mammat et al. ^[2] studied an ecological model with a trophic chain with a classical Lotka-Volterra functional response and found a parameter space where a Hopf bifurcation occurs. Moreover, it is possible to find bifurcation points analytically and to show that the system has periodic solutions around these points. Rihand et al. ^[3] studied the dynamics of a two-prey one-predator system, where the growth of both prey populations is subject to Allee effects, and there is a direct competition between them. Castillo et al. ^[4] demonstrated the existence of a limit cycle, via the first Lyapunov coefficient and the Andronov-Hopf bifurcation theorem, for an asymmetric intragremial food web model with Holling type Ⅱ functional response for intermediate and top predators and logistic growth for (common) prey. Cheng et al. ^[5] reviewed the existence and local stabilities of all equilibria of the classical Holling type Ⅱ model of the three-species food chain. They obtained the existence of a single limit cycle when the limit equilibrium loses its stability, and they also demonstrated the global stability of the limit cycle in $\mathbb{R}^3$ . Alsakaji et al. ^[6] studied the dynamics of a delay differential model of predator-prey system involving teams of two-prey and one-predator, with Monod-Haldane and Holling type Ⅱ functional responses, and a cooperation between the two-teams of preys against predation. Blé et al. ^[7] demonstrated that a tritrophic chain model with Holling type Ⅲ functional response has an equilibrium point where it presents a supercritical Hopf bifurcation independently of the prey growth rate. In the logistic case, they demonstrate the existence of at least three equilibrium points in the positive octant and one of them presents a supercritical Hopf bifurcation.

On the other hand, the theory of fractional order differential equations has gained great popularity in several scientific areas; such as biomathematics, control theory, and financial mathematics, among others (see ^[8,9,10,11]). In particular, in the case of biomathematics, tritrophic models with fractional derivative have been studied; for example, Maria et al. ^[12] calculated the equilibrium points and analyzed their stability to exhibit the dynamic behavior of a fractional order prey-predator model (3-Species). Mondal et al. ^[13] studied the dynamics of a three-dimensional discrete fractional-order eco-epidemiological model with Holling type Ⅱ functional response. They determined analytical conditions for the local stability of different fixed points using the Jury criterion and showed that the stability of the fractional-order discrete system depends strongly on the step size and the fractional order. More specifically, the critical value of the step size, at which stability switching occurs, decreases as the order of the fractional derivative decreases.

In the literature, we did not find results concerning the study of fractional order tritrophic systems with Holling type Ⅲ functional response.

In this work, we study the stability for the positive solution of the fractional tritrophic food chain models:

$\begin{eqnarray} {D}_{t_{0}}^{\alpha}x& = &\rho x-k_{1}f_{i}(x)y, \\ {D}_{t_{0}}^{\alpha}y& = &c_{1}f_{i}(x)y-k_{2}g_{i}(y)z-c_{2}y,\\ {D}_{t_{0}}^{\alpha}z& = &c_{3}g_{i}(y)z-dz, \end{eqnarray}$

(1.1)

where $x$ , $y$ , $z$ represent the density of the prey, mesopredator, and superpredator, respectively; ${D}_{t_{0}}^{\alpha}$ is the Caputo fractional derivative, $\alpha$ $\in (0, 1]$ , and all parameters are nonnegative. The parameters $c_1$ and $c_3$ represent the benefits of food consumption, $k_1$ and $k_2$ are the predator rates of the mesopredator, and superpredator, respectively, $c_2$ and $d$ are the mortality rate of the corresponding predators, and $\rho$ is the growth rate of the prey in the absence of predators. Since system (1.1) is an ecological model, our region of interest is the positive octant $\Omega = \{(x, y, z)\in \mathbb{R}^3 | x > 0, y > 0, z > 0 \}.$ Also, $f_{2}$ , $g_{2}$ and $f_{3}$ , $g_{3}$ are the types Ⅱ and Ⅲ functional responses, respectively, defined as follows:

$f_{i}(x) = {x^{i-1} \over x^{i-1}+a_{1}}, \ \ g_{i}(y) = {y^{i-1} \over y^{i-1}+a_{2}}, \ \ i = 2,3.$

The type Ⅱ functional response is used to describe predator organisms which take some time capture and ingest their prey. On the other hand, type Ⅲ functional response is characteristic of predator organisms which do not capture prey intensively below a certain level of threshold density; however, above that density level, predator organisms increase their feeding rates until some saturation level is reached.

2. Preliminaries

We begin this section by defining the Hurwitz polynomial as follows.

Definition 2.1. A polynomial $p$ is said to be a Hurwitz polynomial or Hurwitz stable if all the roots $s_{i}$ of $p$ lie in the open left half plane $\mathbb{C}_{-}$ .

We assume that $p(s)$ is the following polynomial:

$\begin{equation} p(s) = A_{0} +A_{1} s + A_{2} s^{2} +A_{3} s^{3}, \end{equation}$

(2.1)

where $A_{i}\in\mathbb{R}$ , $i = 0, 1, 2, 3$ and $A_{3} > 0$ , and $A_{0}\neq 0$ . Using the coefficients of $p(s)$ , we construct the Table 1 (see ^[15]).

Table 1. The Routh table.

$A_{3}$	$A_{1}$	0
$A_{2}$	$A_{0}$	0
$A_{1}-\dfrac{A_{0}A_{3}}{A_{2}}$	0	0
$A_{0}$	0	0

| Show Table

DownLoad: CSV

Theorem 2.1. Consider $p(s)$ given in (2.1). Then, $p(s)$ is Hurwitz if, and only if, each element of the first column of the Routh table is positive, i.e., $A_{3} > 0$ , $A_{2} > 0$ , $A_{1}-\dfrac{A_{0}A_{3}}{A_{2}} > 0$ , and $A_{0} > 0$ .

Proof. The proof can be found in ^[15].

Theorem 2.2. Consider $p(s)$ given in (2.1). Suppose when calculating the Routh table that no element in the first column is zero. Then, the number of sign changes in the first column of the Routh table is the number of open right half-plane zeros of $p(s)$ .

Proof. The proof can be found in ^[15].

Now, we give some definitions and properties of fractional calculus theory.

Definition 2.2. (See ^[11]) Let $x(t)\in L_{1}(\mathbb{R}^{+})$ . The Riemann-Liouville fractional integral of order $\alpha\in (0, 1]$ is defined by

$(I_{t_{0}}^{\alpha}x)(t) = {1\over \Gamma(\alpha)}\int_{t_{0}}^t {x(\tau)\over (t-\tau)^{1-\alpha}} > d\tau, \ \ \ \ t > t_{0}\geq 0,$

where $\Gamma$ is the Gamma function.

Definition 2.3. (See ^[11]) The Caputo fractional derivative of order $\alpha\in (0, 1]$ and $x\in AC(\mathbb{R}^{+})$ , the space of absolutely continuous functions on $\mathbb{R}^{+}$ , is defined by

$(D_{t_{0}}^{\alpha}x)(t) = (I_{t_{0}}^{1-\alpha}Dx)(t),$

where $D = d/dt$ .

The autonomous nonlinear differential system in the sense of Caputo is given as follows:

$\begin{equation} D_{t_{0}}^{\alpha} X(t) = F(X(t)), \end{equation}$

(2.2)

where $X(t)\in \mathbb{R}^3$ , $X(t_{0}) = X_{t_{0}}$ , $F(X)$ is continuous.

Definition 2.4. Suppose that $P$ is an equilibrium point of system (2.2) and that all the eigenvalues $\lambda$ of the linearized matrix $J(P) = \partial F / \partial X |_{X = P}$ evaluated at $P$ satisfy: $|\lambda| \neq 0$ and $|arg(\lambda)|\neq \dfrac{\pi \alpha}{2}$ , then we call $P$ a hyperbolic equilibrium point.

Theorem 2.3. If $P$ is a hyperbolic equilibrium point of (2.2), then vector field $F(x)$ is topologically equivalent with its linearization vector field $J(P)x$ in the neighborhood $\delta (P)$ .

Proof. The proof can be found in ^[16].

Theorem 2.4. We consider the fractional-order system

$\begin{equation} D^{\alpha}X(t) = F(X(t)), \end{equation}$

(2.3)

where $0 < \alpha < 1$ . System (2.3) is asymptotically stable at the equilibrium point $P$ if, and only if,

$| arg(\lambda)| > {\pi \alpha \over 2}$

for all roots $\lambda$ of the following equation $\det(\lambda I - J(P)) = 0$ .

Proof. The proof can be found in ^[17].

3. Main result

In this section, we present two results obtained by analyzing the stability around the coexistence equilibrium points of three species of the tritrophic system (1.1). First, we consider the system (1.1) with Holling type Ⅱ functional response, which has only one unstable equilibrium point in the first octant.

Theorem 3.1. We assume $\alpha = 1$ , $i = 2, \ c_{3} > d, \ a_{2}dk_{1} > a_{1}\rho(c_{3}-d), \ (c_{1}-c_{2})a_{2}dk_{1} > a_{1}c_{1}\rho (c_{3}-d)$ . Then, there is only one equilibrium point $P$ in the first octant for the system (1.1),

$P = \Big( {k_{1}a_{2}d_-\rho a_{1}(c_{3}-d) \over \rho (c_{3}-d)}, {a_{2}d \over c_{3}-d}, {c_{3}((c_{1}-c_{2})k_{1}a_{2}d-\rho a_{1}c_{1}(c_{3}-d))\over dk_{1}k_{2}(c_{3}-d)} \Big).$

Also, the system (1.1) is unstable around the equilibrium point $P$ .

Proof. Let

$\begin{eqnarray} F_{1}(x, y)& = &\rho x-{k_{1}xy \over x+a_{1}}, \\ F_{2}(x, y, z)& = &{c_{1}xy \over x+a_{1}}-{k_{2}yz \over y+a_{2}}-c_{2}y,\\ F_{3}(y, z)& = &{c_{3}yz \over y+a_{2}}-dz. \end{eqnarray}$

(3.1)

The equilibrium point $P \in \Omega$ for system (1.1) when $i = 2$ is obtained by solving system $F_{1}(x, y) = F_{2}(x, y, z) = F_{3}(y, z) = 0$ .

On the other hand, we obtain the Jacobian matrix $J$ of (3.1)

$J = \begin{pmatrix} \rho-\dfrac{a_{1}k_{1}y}{(x+a_{1})^2} & -\dfrac{k_{1}x}{x+a_{1}} & 0 \\ \dfrac{a_{1}c_{1}y}{(x+a_{1})^2} & \dfrac{c_{1}x}{x+a_{1}}-\dfrac{a_{2}k_{2}z}{(y+a_{2})^2}-c_{2} & \dfrac{-k_{2}y}{y+a_{2}} \\ 0 & \dfrac{a_{2}c_{3}z}{(y+a_{2})^2} & \dfrac{c_{3}y}{y+a_{2}}-d \end{pmatrix}.$

Now, evaluate $P$ in $J$

$J(P)$ =

$\begin{pmatrix} \dfrac{\rho(a_{2}dk_{1}-a_{1}\rho(c_{3}-d))}{a_{2}dk_{1}} & \dfrac{\rho a_{1}(c_{3}-d)-k_{1}a_{2}d}{a_{2}d} & 0 \\ \dfrac{a_{1}c_{1}\rho^{2}(c_{3}-d)}{a_{2}dk_{1}^{2}} & \dfrac{a_{2}dk_{1}(c_{1}-c_{2})-a_{1}c_{1}\rho (c_{3}-d)}{ a_{2}c_{3}k_{1}} & \dfrac{-dk_{2}}{ c_{3}} \\ 0 & \dfrac{(c_{3}-d)((c_{1}-c_{2})a_{2}dk_{1}-a_{1}c_{1}\rho (c_{3}-d))}{ a_{2}dk_{1}k_{2}} & 0 \end{pmatrix}$ .

We denote by $J(P) = (m_{jk})$ , $j, k = 1, 2, 3$ . The characteristic polynomial is

$\begin{equation} p(\lambda) = \det[J(P)-\lambda I], \end{equation}$

(3.2)

where $I$ is the identity matrix 3x3. That is,

$\begin{equation} p(\lambda) = A_3 \lambda^{3}+A_2 \lambda^{2}+A_1 \lambda+A_0, \end{equation}$

(3.3)

with

$A_3 = 1, \ A_2 = -m_{11}-m_{22}, \ A_1 = m_{11}m_{22}-m_{12}m_{21}-m_{23}m_{32}, \ A_0 = m_{11}m_{23}m_{32}.$

Let $P = (x_0, y_0, z_0)$ be the equilibrium point in the first octant. We reduce $m_{11}$ and $m_{32}$ :

$m_{11} = \frac{\rho^2 x_{0}}{k_{1}y_{0}}, \ \ m_{32} = \frac{a_{2}d^2 z_{0}}{c_{3}y_{0}^2}.$

Then

$A_{0} = - \frac{a_{2}d^{3}k_{2}\rho^{2}x_{0}z_{0}}{c_{3}^{2}k_{1}y_{0}^{3}} < 0.$

Therefore, by Theorem 2.2, the system is unstable.

Now, we consider the system (1.1) with Holling type Ⅲ functional response, which has two equilibrium points in the first octant, one stable and the other unstable.

Theorem 3.2. Assuming $i = 3, \ k_{1} = 2\rho \sqrt{r}$ , $c_{3} = rd$ , $c_{1} = rc_{2}$ , $a_{1} = a_{2}$ , and $r > 4$ , the system (1.1) has points of equilibrium in the first octant:

$\begin{eqnarray*} P_{1}& = &\bigg({ \sqrt{a_{1}} (\sqrt{r}+1) \over \sqrt{r-1}}, \sqrt{{a_{1} \over r-1}}, { \sqrt{a_{1}} c_{2} r (r+\sqrt{r}-2) \over 2 k_{2} \sqrt{r-1}}\bigg),\\ P_{2}& = &\bigg({ \sqrt{a_{1}} (\sqrt{r}-1) \over \sqrt{r-1}}, \sqrt{{a_{1} \over r-1}}, { \sqrt{a_{1}} c_{2} r (r-\sqrt{r}-2) \over 2 k_{2} \sqrt{r-1}}\bigg). \end{eqnarray*}$

Then, the system (1.1) is unstable around the equilibrium point $P_{1}$ and asymptotically stable around $P_{2}$ .

Proof. Let

$\begin{eqnarray} G_{1}(x, y)& = &\rho x-{2\rho \sqrt{r} x^{2}y \over x^2 +a_{1}}, \\ G_{2}(x, y, z)& = &{c_{2}rx^2 y \over x^2 +a_{1}}-{k_{2}y^2 z \over y^2 +a_{1}}-c_{2}y,\\ G_{3}(y, z)& = &{dry^2 z \over y^2 +a_{1}}-dz. \end{eqnarray}$

(3.4)

Then, the Jacobian matrix $J$ of system (3.4) is

$J$ =

$\begin{pmatrix} \rho-\dfrac{4 a_{1} \rho \sqrt{r} x y}{(x^2 +a_{1})^2} & \dfrac{-2 \rho \sqrt{r}x^2}{ x^2 +a_{1}} & 0 \\ \dfrac{2 a_{1}c_{2} xy}{(x^2 +a_{1})^2} & \dfrac{c_{2}rx^2}{ x^2 +a_{1}}-\dfrac{2a_{1}k_{2}yz }{(y^2 +a_{1})^2}-c_{2} & \dfrac{-k_{2}y^2}{ y^2 +a_{1}} \\ 0 & \dfrac{2a_{1}dryz}{(y^2 +a_{1})^2} & \dfrac{dry^2}{y^2 +a_{1}}-d \end{pmatrix}$ .

We evaluate $J$ in $P_{1}$ ,

$J(P_{1})$ =

$\begin{pmatrix} \dfrac{\rho }{\sqrt{r}} & -\rho (\sqrt{r}+1) & 0 \\ \dfrac{c_{2} (\sqrt{r}-1)}{ 2} & c_{2}\Big(-\dfrac{r }{ 2}-\dfrac{\sqrt{r} }{ 2}+\dfrac{1 }{ \sqrt{r}}-\dfrac{2 }{ r}+2 \Big) & -\dfrac{k_{2} }{ r} \\ 0 & \dfrac{c_{2}d }{ k_{2}} \Big( r^2 +r^{3/2}-3r-\sqrt{r}+2\Big) & 0 \end{pmatrix}$ .

Let $J(P_{1}) = (m_{jk})$ , $j, k = 1, 2, 3$ . We calculate the characteristic polynomial:

$p(\lambda) = \det[J(P_{1})-\lambda I].$

That is,

$\begin{equation} p(\lambda) = A_{3} \lambda^{3}+A_{2} \lambda^{2}+A_{1} \lambda+A_{0}, \end{equation}$

(3.5)

where

$\begin{eqnarray*} A_{0}& = &-{c_{2}d \rho \over r^{3/2}} (r-1)(r+\sqrt{r}-2),\\ A_{1}& = &{c_{2}\rho \over \sqrt{r}}\Big( {r^{3/2} \over 2}-\frac{r}{2}-\sqrt{r}+\frac{1}{\sqrt{r}}-\frac{2}{r}+2\Big)+{c_{2}d \over r} \Big( r^2 +r^{3/2}-3r-\sqrt{r}+2\Big),\\ A_{2}& = &c_{2}\Big({r \over 2}+{\sqrt{r} \over 2}-2-{1 \over \sqrt{r}}+{2 \over r} \Big)-{\rho \over \sqrt{r}},\\ A_{3}& = &1. \end{eqnarray*}$

We note that $A_{0} < 0$ , when $r > 4$ . Then, by Theorem 2.1, $p(\lambda)$ is not a Hurwitz polynomial.

On the other hand,

$J(P_{2})$ =

$\begin{pmatrix} -\dfrac{\rho }{ \sqrt{r}} & -\rho (\sqrt{r}-1) & 0 \\ \dfrac{c_{2} (\sqrt{r}+1)}{ 2} & c_{2}\Big(-\dfrac{r }{ 2}+\dfrac{\sqrt{r} }{ 2}+2-\dfrac{1 }{ \sqrt{r}}-\dfrac{2}{ r} \Big) & -\dfrac{k_{2} }{ r} \\ 0 & \dfrac{c_{2}d }{ k_{2}} \Big( r^2 -r^{3/2}-3r+\sqrt{r}+2\Big) & 0 \end{pmatrix}$ .

The characteristic polynomial has the form (3.5), where $A_{i}$ are

$\begin{eqnarray*} A_{0}& = &{c_{2}d \rho \over r^{3/2}} (r-1)(r-\sqrt{r}-2),\\ A_{1}& = &{c_{2}\rho \over \sqrt{r}}\Big( {r^{3/2} \over 2}+\frac{r}{2}-\sqrt{r}-2+\frac{1}{\sqrt{r}}+\frac{2}{r}\Big)+{c_{2}d \over r} (r-1)(r-\sqrt{r}-2), \\ A_{2}& = &c_{2}\Big({r \over 2}-{\sqrt{r} \over 2}-2+{1 \over \sqrt{r}}+{2 \over r} \Big)+{\rho \over \sqrt{r}},\\ A_{3}& = &1. \end{eqnarray*}$

Also, $r > 4$ , from which we obtain

$\begin{eqnarray} & & r-\sqrt{r}-2 > 0, \\ \dfrac{r^{3/2}}{2}+\dfrac{r}{2}-\sqrt{r}-2+\dfrac{1}{\sqrt{r}}+\dfrac{2}{r}& > &{r \over 2}-{\sqrt{r} \over 2}-2+{1 \over \sqrt{r}}+{2 \over r} > \biggr({\sqrt{r} \over 2}-{2 \over \sqrt{r}} \biggr)^{2}+\dfrac{r}{4}-\dfrac{\sqrt{r}}{2} > 0. \end{eqnarray}$

(3.6)

For (3.6), we have $A_{i} > 0$ , for each $i = 0, 1, 2$ . Now, we calculate $A_{1}A_{2}-A_{0}$ .

$\begin{equation*} A_{1}A_{2}-A_{0} = {c_{2}\rho A_{1} \over \sqrt{r}}\Big( {r^{3/2} \over 2}+\frac{r}{2}-\sqrt{r}-2+\frac{1}{\sqrt{r}}+\frac{2}{r}\Big)+{c_{2}^{2}d \over r} (r-1)(r-\sqrt{r}-2)\Big({r \over 2}-{\sqrt{r} \over 2}-2+{1 \over \sqrt{r}}+{2 \over r} \Big). \end{equation*}$

By (3.6) and $A_{1} > 0$ , we have $A_{1}A_{2}-A_{0} > 0$ . Then, by Theorem 2.1, $p(\lambda)$ is a Hurwitz polynomial. Therefore, the system (1.1), when $\alpha = 1$ , is stable around $P_2$ .

Now, let us consider $\alpha \in (0, 1)$ . By Theorem 2.3, the system (1.1) and its linearization vector field $J(P_{2})X$ are topologically equivalent. By Theorem 2.4, $D^{\alpha}X = J(P_{2})X$ is stable around $P_2$ . Therefore, the system (1.1) is stable around $P_2$ .

4. Analytical solution

In this section, we present the construction of the analytical solution for the system (1.1), with Holling type Ⅲ functional response, $k_{1} = 2\rho \sqrt{r}$ , $c_{3} = rd$ , $c_{1} = rc_{2}$ , and $a_{1} = a_{2}$ , which are obtained by applying the multistage homotopy perturbation method (see ^[18]):

First, let's define a regular partition of the interval $[0, T]$ by $t_{0} = 0 < t_{1} < t_{2} < \dots < t_{m} = T$ and the following family of fractional-order systems

$\begin{eqnarray} & &{D}_{t_{k-1}}^{\alpha}x_k = \rho x_k-{2\rho \sqrt{r}x_{k}^{2}y_k \over x_k^{2}+a_{1}},\\ & & {D}_{t_{k-1}}^{\alpha}y_k = \frac{c_{2} r x_{k}^2 y_k}{x_{k}^2 +a_{1}}-{k_{2}y_k^{2}z_k \over y_k^{2}+a_{1}}-c_{2}y_{k},\\ & & {D}_{t_{k-1}}^{\alpha}z_k = \frac{dr y_{k}^2 z_{k}}{y_{k}^2 +a_{1}}-dz_{k}, \end{eqnarray}$

(4.1)

with $t \in [t_{k-1}, t_{k}]$ and initial conditions $(x_{0}, y_{0}, z_{0}) = (x_{1}(t_0), y_{1}(t_0), y_{1}(t_0))$ and $(x_k(t_{k-1}), y_k(t_{k-1}), z_k(t_{k-1})) = (x_{k-1}(t_{k-1}), y_{k-1}(t_{k-1}), z_{k-1}(t_{k-1}))$ for $k = 1, \dots, m$ .

Now, we define the homotopy for each $k$ :

$\begin{eqnarray} & &(1-p)\big({D}_{t_{k-1}}^{\alpha}u_k-{D}_{t_{k-1}}^{\alpha}x_{k}(t_{k-1})\big) +p\bigg({D}_{t_{k-1}}^{\alpha}u_k- \rho u_k+{2\rho \sqrt{r}u_{k}^{2}v_k \over u_k^{2}+a_{1}}\bigg) = 0, \\ & &(1-p)\big({D}_{t_{k-1}}^{\alpha}v_k-{D}_{t_{k-1}}^{\alpha}y_{k}(t_{k-1})\big) +p\bigg({D}_{t_{k-1}}^{\alpha}v_k- \frac{c_{2} r u_{k}^2 v_k}{x_{k}^2 +a_{1}}+{k_{2}v_k^{2}w_k \over v_k^{2}+a_{1}}+c_{2}v_{k}\bigg) = 0,\\ & &(1-p)\big({D}_{t_{k-1}}^{\alpha}w_k-{D}_{t_{k-1}}^{\alpha}z_{k}(t_{k-1})\big) +p\bigg({D}_{t_{k-1}}^{\alpha}w_k+cw_k-{dv_k^{2} w_k \over v_k^{2}+e}\bigg) = 0, \end{eqnarray}$

(4.2)

where $p\in[0, 1]$ . Let's suppose that the solution for (4.2) is given by

$\begin{eqnarray} u_{k}& = &u_{0k}+pu_{1k}+p^{2}u_{2k}+p^{3}u_{3k}+\cdots, \\ v_{k}& = &v_{0k}+pv_{1k}+p^{2}v_{2k}+p^{3}v_{3k}+\cdots,\\ w_{k}& = &w_{0k}+pw_{1k}+p^{2}w_{2k}+p^{3}w_{3k}+\cdots, \end{eqnarray}$

(4.3)

where $u_{ik}$ , $v_{ik}$ , $w_{ik}$ , $i = 1, 2, \dots$ , are functions to be determined. Then, the solution for $t\in[t_{k-1}, t_{k}]$ is

$\begin{eqnarray} x_{k} = \lim\limits_{p \rightarrow 1 } u_{k}& = &u_{0k}+u_{1k}+u_{2k}+u_{3k}+\cdots, \\ y_{k} = \lim\limits_{ p \rightarrow 1 }v_{k}& = &v_{0k}+v_{1k}+v_{2k}+v_{3k}+\cdots,\\ z_{k} = \lim\limits_{ p \rightarrow 1 }w_{k}& = &w_{0k}+w_{1k}+w_{2k}+w_{3k}+\cdots. \end{eqnarray}$

(4.4)

Therefore, the analytic solution for $t\in [0, T]$ is given by

$\begin{eqnarray} x(t)& = &\sum\limits_{k = 1}^{m}I_{[t_{k-1},t_k]} \sum\limits_{j = 1}^{\infty} u_{jk}(t), \\ y(t)& = &\sum\limits_{k = 1}^{m}I_{[t_{k-1},t_k]} \sum\limits_{j = 1}^{\infty} v_{jk}(t), \\ z(t)& = &\sum\limits_{k = 1}^{m}I_{[t_{k-1},t_k]} \sum\limits_{j = 1}^{\infty} w_{jk}(t), \end{eqnarray}$

(4.5)

where $I_{[t_{k-1}, t_k]}$ is the characteristic function, $t_{k} = k\dfrac{T}{m}$ , and $\bigcup\limits_{k = 1}^{m} [t_{k-1}, t_k] = [0, T]$ . We show the first addends of the series (4.4),

$\begin{eqnarray*} u_{1k}(t)& = & \dfrac{\rho u_{0k} (a_{1} + u_{0k}(u_{0k} - 2 \sqrt{r} v_{0k}))}{(a_{1}+ u_{0k}^2)\Gamma(1 + \alpha)}(t-t_{k-1} )^{\alpha}, \nonumber \\ v_{1k}(t)& = & - \dfrac{v_{0k}(c_{2}(a_{1}-(r-1)u_{0k}^2)(a_{1}+v_{0k}^2) +k_{2}(a_{1}+u_{0k}^2)v_{0k}w_{0k})}{\alpha (a_{1}+u_{0k}^2)(a_{1}+v_{0k}^2)\Gamma(\alpha)}(t-t_{k-1} )^{\alpha}, \\ w_{1k}(t)& = & - \dfrac{d w_{0k}(a_{1}-(r-1)v_{0k}^2)}{(a_{1}+v_{0k}^2)\Gamma(1+\alpha)}(t-t_{k-1} )^{\alpha}, \nonumber \\ u_{2k}(t)& = & \dfrac{\rho u_{0k}}{(a_{1}+u_{0k}^2)^{3}\Gamma^2 (1+\alpha)}(a_{1}\rho (a_{1}+u_{0k}^2)(a_{1}+u_{0k}(u_{0k}-2\sqrt{r}v_{0k}))+3\rho u_{0k}^2 (a_{1}+u_{0k}^2)(a_{1}+u_{0k} \nonumber \\ & & \times (u_{0k} -2\sqrt{r}v_{0k}))-4\rho \sqrt{r}u_{0k}v_{0k}(a_{1}+u_{0k}^2)(a_{1}+u_{0k}(u_{0k}-2\sqrt{r}v_{0k}))-2\rho u_{0k}^2 (a_{1}+u_{0k} \nonumber \\ & & \times (u_{0k}-2\sqrt{r}v_{0k}))^2+\dfrac{2\sqrt{r}u_{0k}v_{0k}}{a_{1}+v_{0k}^2}(a_{1}+u_{0k}^2)(c_{2}(a_{1}-(r-1)u_{0k}^2)(a_{1}+v_{0k}^2)+k_{2}(a_{1}+u_{0k}^2) \nonumber \\ & & \times v_{0k}w_{0k}))(t-t_{k-1} )^{2\alpha}, \nonumber \\ v_{2k}(t)& = &\dfrac{v_{0k}}{(a_{1}+u_{0k}^2 )^3 (a_{1}+v_{0k}^2 )^3 \Gamma^2 (1+\alpha)}((a_{1}+u_{0k}^2 )((a_{1}+v_{0k}^2 )c_{2}(a_{1}-(r-1)u_{0k}^2 )+k_{2}(a_{1}+u_{0k}^2 ) \nonumber \\ & &\times v_{0k}w_{0k})(c_{2}(a_{1}-(r-1)u_{0k}^2 )(a_{1}+v_{0k}^2 )^2 +2a_{1}k_{2}(a_{1}+u_{0k}^2 )v_{0k}w_{0k})+(a_{1}+v_{0k}^2 )(2a_{1}c_{2}\rho \nonumber \\ & & \times u_{0k}^2 r(a_{1}+ v_{0k}^2 )^2(a_{1}+u_{0k}(u_{0k}-2\sqrt{r}v_{0k}))+dk_{2}(a_{1}+u_{0k}^2 )^3 v_{0k}(a_{1}-(r-1)v_{0k}^2 )w_{0k})) \nonumber \\ & & \times(t-t_{k-1} )^{2\alpha}, \nonumber \\ w_{2k}(t)& = &\dfrac{dw_{0k}}{(a_{1}+u_{0k}^2 )(a_{1}+v_{0k}^2 )^3 \alpha \Gamma^2 (1+\alpha)}(d (a_1 + u_{0k}^2) (a_1 + v_{0k}^2 ) (a_{1} - (r-1) v_{0k}^2 )^2 \alpha - 2 a_1 r v_{0k}^2 (c_2 \nonumber \\ & & \times (a_1 - (r-1) u_{0k}^2) (a_1 + v_{0k}^2 ) + k_2 (a_1 + u_{0k}^2 ) v_{k0} w_{0k} ) \alpha)(t-t_{k-1})^{2\alpha}. \nonumber \end{eqnarray*}$

5. Examples

In this section, we illustrate the results obtained in Theorems 3.1 and 3.2, through some particular examples, where its graphs show the dynamics of the solutions around the equilibrium points. Furthermore, we can observe that as the order of the derivative $\alpha$ moves away from $1$ , the length of the trajectories increases. For this purpose, we use the analytical solution given in (4.5).

Example 5.1. (See ) For $i = 2$ , fix $a_{1} = c_{2} = \rho = r = 1$ , $c_{1} = d = k_{1} = k_{2} = 2$ , $c_{3} = 2.5$ , $a_{2} = 5$ , $m = 50$ , $t \in [0, 0.5]$ , $(16, 8, 5)$ is the initial condition, and $P = (39, 20, 11.875)$ is an unstable equilibrium point.

Figure 1. Unstable equilibrium point

$P$ .

DownLoad: Full-Size Img PowerPoint

Example 5.2. (See ) For $i = 3$ :

Figure 2. Unstable equilibrium point

$P_{1}$ and asymptotic stability around

$P_{2}$ .

DownLoad: Full-Size Img PowerPoint

● Let $a_{1} = c_{2} = d = k_{2} = \rho = r = 1$ , $r = 5$ , $m = 100$ , $t \in [0, 1]$ , $(2, 1, 7)$ is the initial condition, and $P_{1} = (1.61803, 0.5, 6.54508)$ is an unstable equilibrium point.

● We take $a_{1} = c_{2} = d = k_{2} = \rho = r = 5$ , $m = 100$ , $t \in [0, 1]$ , $(1, 1, 2)$ is the initial condition, and $P_{2} = (1.38197, 1.11803, 2.13525)$ is a locally asymptotically stable equilibrium point.

6. Conclusions

In this work, the local stability around the coexistence equilibrium points of two tritrophic fractional models were studied, with Holling type Ⅱ and Ⅲ functional responses, respectively. Under certain conditions of the parameters, it was found that only the second model has a stable equilibrium point $P_2$ , when $\alpha \in (0, 1]$ . The multistage homotopic perturbation method was used to obtain the solution of the tritrophic fractional model with Holling type Ⅲ function response, and it was observed that the trajectories tend faster toward the equilibrium point when $\alpha < 1$ . These results reveal the possibility of tritrophic coexistence when the interaction is Holling type Ⅲ. A future work is to use the analytical solution, obtained via the homotopic perturbation method, to solve the inverse problem associated to system (1.1); that is, to estimate the parameters involved in the system, from a Bayesian analysis perspective, which is important to model experimental problems in ecology.

Author contributions

Anel Esquivel-Navarrete: investigation, formal analysis, writing – review $\And$ editing; Jorge Sanchez-Ortiz: investigation, supervision, writing – review $\And$ editing; Gabriel Catalan-Angeles: formal analysis, writing- original draft preparation, visualization; Martin P. Arciga-Alejandre: investigation, methodology, writing – review $\And$ editing.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare there is no conflict of interest.

References

[1]	M. Alfaro, J. Coville, Propagation phenomena in monostable integro-differential equations: Acceleration or not?, J. Differ. Equations, 263 (2017), 5727–5758. https://doi.org/10.1016/j.jde.2017.06.035 doi: 10.1016/j.jde.2017.06.035
[2]	F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi, J. Toledo-Melero, Nonlocal diffusion problems, Providence, Rhode Island: AMS, 2010. https://doi.org/10.1090/surv/165
[3]	P. W. Bates, G. Zhao, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl., 332 (2007), 428–440. https://doi.org/10.1016/j.jmaa.2006.09.007 doi: 10.1016/j.jmaa.2006.09.007
[4]	H. Berestycki, J. Coville, H. Vo, On the definition and the properties of the principal eigenvalue of some nonlocal operators, J. Funct. Anal., 271 (2016), 2701–2751. https://doi.org/10.1016/j.jfa.2016.05.017 doi: 10.1016/j.jfa.2016.05.017
[5]	E. Bouin, J. Garnier, C. Henderson, F. Patout, Thin front limit of an integro-differential Fisher-KPP equation with fat-tailed kernels, SIAM J. Math. Anal., 50 (2018), 3365–3394. https://doi.org/10.1137/17M1132501 doi: 10.1137/17M1132501
[6]	X. Cabré, J.-M. Roquejoffre, The influence of fractional diffusion in Fisher-KPP equations, Commun. Math. Phys., 320 (2013), 679–722. https://doi.org/10.1007/s00220-013-1682-5 doi: 10.1007/s00220-013-1682-5
[7]	J. Cao, Y. Du, F. Li, W. T. Li, The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries, J. Funct. Anal., 277 (2019), 2772–2814. https://doi.org/10.1016/j.jfa.2019.02.013 doi: 10.1016/j.jfa.2019.02.013
[8]	C. Cortázar, F. Quirós, N. Wolanski, A nonlocal diffusion problem with a sharp free boundary, Interface. Free Bound., 21 (2019), 441–462. https://doi.org/10.4171/ifb/430 doi: 10.4171/ifb/430
[9]	J. Coville, L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinb. A, 137 (2007), 727–755. https://doi.org/10.1017/S0308210504000721 doi: 10.1017/S0308210504000721
[10]	J. Coville, J. Davila, S. Martinez, Pulsating fronts for nonlocal dispersion and KPP nonlinearity, Ann. Inst. Henri Poincaré (C) Non Linear Anal., 30 (2013), 179–223. https://doi.org/10.1016/j.anihpc.2012.07.005 doi: 10.1016/j.anihpc.2012.07.005
[11]	F. del Teso, J. Endal, J. L. Vazquez, The one-phase fractional Stefan problem, Math. Mod. Meth. Appl. Sci., 31 (2021), 83–131. https://doi.org/10.1142/S0218202521500032 doi: 10.1142/S0218202521500032
[12]	Y. Du, Z. Guo, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary Ⅱ, J. Differ. Equations, 250 (2011), 4336–4366. https://doi.org/10.1016/j.jde.2011.02.011 doi: 10.1016/j.jde.2011.02.011
[13]	Y. Du, Z. Guo, The Stefan problem for the Fisher-KPP equation, J. Differ. Equations, 253 (2012), 996–1035. https://doi.org/10.1016/j.jde.2012.04.014 doi: 10.1016/j.jde.2012.04.014
[14]	Y. Du, F. Li, M. Zhou, Semi-wave and spreading speed of the nonlocal Fisher-KPP equation with free boundaries, J. Math. Pure Appl., 154 (2021), 30–66. https://doi.org/10.1016/j.matpur.2021.08.008 doi: 10.1016/j.matpur.2021.08.008
[15]	Y. Du, Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377–405. https://doi.org/10.1137/090771089 doi: 10.1137/090771089
[16]	Y. Du, H. Matsuzawa, M. Zhou, Spreading speed and profile for nonlinear Stefan problems in high space dimensions, J. Math. Pure. Appl., 103 (2015), 741–787. https://doi.org/10.1016/j.matpur.2014.07.008 doi: 10.1016/j.matpur.2014.07.008
[17]	Y. Du, W. Ni, Rate of propagation for the Fisher-KPP equation with nonlocal diffusion and free boundaries, 2021, preprint. Available from: http://turing.une.edu.au/ydu/papers/SpreadingRate-July2021.pdf
[18]	Y. Du, W. Ni, The high dimensional Fisher-KPP nonlocal diffusion equation with free boundary and radial symmetry, Part 1, SIAM J. Math. Anal., in press.
[19]	Y. Du, W. Ni, The high dimensional Fisher-KPP nonlocal diffusion equation with free boundary and radial symmetry: sharp estimates, 2022, preprint. Available from: http://turing.une.edu.au/ydu/papers/dn-highD-2-March2022.pdf
[20]	J. Fang, G. Faye, Monotone traveling waves for delayed neural field equations, Math. Mod. Meth. Appl. Sci., 26 (2016), 1919–1954. https://doi.org/10.1142/S0218202516500482 doi: 10.1142/S0218202516500482
[21]	D. Finkelshtein, P. Tkachov, Accelerated nonlocal nonsymmetric dispersion for monostable equations on the real line, Appl. Anal., 98 (2019), 756–780. https://doi.org/10.1080/00036811.2017.1400537 doi: 10.1080/00036811.2017.1400537
[22]	R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 355–369. https://doi.org/10.1111/j.1469-1809.1937.tb02153.x doi: 10.1111/j.1469-1809.1937.tb02153.x
[23]	J. Garnier, Accelerating solutions in integro-differential equations, SIAM J. Math. Anal., 43 (2011), 1955–1974. https://doi.org/10.1137/10080693X doi: 10.1137/10080693X
[24]	A. N. Kolmogorov, I. G. Petrovski, N. S. Piskunov, A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem, Byul. Moskovskogo Gos. Univ., 1 (1937), 1–25.
[25]	F. Li, J. Coville, X. Wang, On eigenvalue problems arising from nonlocal diffusion models, Discrete Contin. Dyn. Syst., 37 (2017), 879–903. https://doi.org/10.3934/dcds.2017036 doi: 10.3934/dcds.2017036
[26]	X. Liang, T. Zhou, Spreading speeds of nonlocal KPP equations in almost periodic media, J. Funct. Anal., 279 (2020), 108723. https://doi.org/10.1016/j.jfa.2020.108723 doi: 10.1016/j.jfa.2020.108723
[27]	W. Shen, A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differ. Equations, 249 (2010), 747–795. https://doi.org/10.1016/j.jde.2010.04.012 doi: 10.1016/j.jde.2010.04.012
[28]	P. E. Souganidis, A. Tarfulea, Front propagation for integro-differential KPP reaction-diffusion equations in periodic media, Nonlinear Differ. Equ. Appl., 26 (2019), 29. https://doi.org/10.1007/s00030-019-0573-7 doi: 10.1007/s00030-019-0573-7
[29]	H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal. 13 (1982), 353–396. https://doi.org/10.1137/0513028 doi: 10.1137/0513028
[30]	H. F. Weinberger, X. Q. Zhao, An extension of the formula for spreading speeds, Math. Biosci. Eng., 7 (2010), 187–194. https://doi.org/10.3934/mbe.2010.7.187 doi: 10.3934/mbe.2010.7.187
[31]	H. Yagisita, Existence and nonexistence of traveling waves for a nonlocal monostable equation, Publ. Res. Inst. Math. Sci., 45 (2009), 925–953. https://doi.org/10.2977/prims/1260476648 doi: 10.2977/prims/1260476648

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Mathematics in Engineering

The Fisher-KPP nonlocal diffusion equation with free boundary and radial symmetry in ${\mathbb R}^3$

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Abstract

1. Introduction

2. Preliminaries

3. Main result

4. Analytical solution

5. Examples

6. Conclusions

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The Fisher-KPP nonlocal diffusion equation with free boundary and radial symmetry in R3 {\mathbb R}^3

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1. Introduction

2. Preliminaries

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5. Examples

6. Conclusions

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The Fisher-KPP nonlocal diffusion equation with free boundary and radial symmetry in ${\mathbb R}^3$