Elimination, permanence, and exclusion in a competition model under Allee effects

1.   Introduction

In traditional methods of estimating population growth rates, people consider that most species reproduce and die proportionally to the current population size. A simple representation of this concept can be expressed with the exponential growth of the populations. There are at least two major reasons to further modify the system: 1) Infinite growth is impossible due to limitation of habitat and resources; 2) Competition between various species should be addressed. We can now introduce the Lotka-Volterra system to address inter-species competition, where $u(t)$ and $v(t)$ represent the population size of two biological species, $b_u$ and $b_v$ are intrinsic birth rates, $d_u$ and $d_v$ are death rates. Throughout this paper we will assume that for all species considered the intrinsic birth rate is larger than death rate. This type of competition, known as Lotka-Volterra competition ^[2,5], implies that species $v$ diminishes resources available to $u$ by a factor of $\alpha$, and $u$ diminishes resources available to $v$ by the factor $\beta$.

Due to the possibility that small population size can make a species vulnerable to extinction, further modifications can be made to the above equations. The Allee effect (given as early as 1932, by Allee ^[1]) models this possibility and was introduced to the multi-species models in recent years ^{[9,14,17,18,20,21]}.

where $k_u$ and $k_v$ are called half-saturation constants. We see that, for large population sizes ($u$ and $v$ much larger than $k_u$ and $k_v$ respectively) the Allee term, $\Big(\frac{u}{u+k_u}\Big)$ or $\Big(\frac{v}{v+k_v}\Big)$, approaches $1$ and has little effect on the modeled change in population. As population size $u$ or $v$ approaches $0$, the Allee effect term gets near $0$ and the death rate ($d_u$ or $d_v$) dominates the differential equation, which drives a small population toward extinction.

We now have the above system of ordinary differential equations that addresses total population sizes, boundedness, and the vulnerability of species for sizes getting too small, as well as the competitive nature of the two species. Our final extension of the system (1.4) is to include the diffusion of both populations, local density and migration, as well as boundary conditions on the habitat. This leads to the following reaction-diffusion system for density functions $u(t, x)$ and $v(t, x)$ with respective diffusion rates $D_u$, $D_v > 0$ for both species:

Here the boundary conditions are given as

with $\nu$ as the normal vector on $\partial \Omega$, $\gamma_u$ and $\gamma_v \in C^{1+\alpha}(\partial \Omega)$, and $\gamma_u(x)$, $\gamma_v(x) \geq 0$ on $\partial \Omega$. This way, we include three commonly used types of boundary conditions: Dirichlet, Neumann, and Robin types.

The above reaction-diffusion system (with one-side Allee effect) on infinite spatial domain was recently studied in ^[9] where the asymptotic stability of the equilibria are given and the existence of traveling wave solutions are proven. In research studies on reaction-diffusion systems modeling multi-species population dynamics in bounded spatial domain (competition, predator-prey, food chain, etc.), much attention has been given to extinction, permanence, and competition or predation caused exclusion ^{[3,6,7,10,11,12,13,15,17,19]}. These discussions are also extended to many variations of the Lotka-Volterra models under different boundary conditions ^[4,8,9]. In this paper, we apply the method of upper-lower solutions to the reaction-diffusion system 1.5 and find conditions for the competing species to self-eliminate (because of Allee effect), coexist (with balanced biological parameters and initial density functions), or competitively exclude (through resource competition and Allee effect). It is seen that the ultimate outcomes in the biological system also depend on the size of the habitat and boundary conditions for both species. According to the theoretical results on permanence and competitive exclusion, we will demonstrate numerical simulations under parameters satisfying conditions obtained.

Examining the reaction functions in system (1.5) by taking the partial derivative of

with respect to v, and the partial derivative of

with respect to u, we find that (for $u$ and $v \geq 0$):

As defined in (^[15], page 383), the system (1.5) is quasi-monotone non-increasing. The upper-lower solutions $(\tilde{u}, \tilde{v})$ and $(\hat{u}, \hat{v})$ defined as following ensure the existence-comparison result given in the below lemma.

Definition 1.1 Upper and lower solutions ^[16].

A pair of smooth functions in $\tilde{U} = (\tilde{u}, \tilde{v})$, and $\hat{U} = (\hat{u}, \hat{v})$ in $C ((0, \infty)\times \bar{\Omega}) \cap C^{1, 2} ((0, \infty)\times\Omega)$ are ordered upper and lower solutions of system (1.5) if they satisfy the relation $\tilde{U} \geq \hat{U}$ and if:

Lemma 1.2. Existence and comparison ^[16].

If there are a pair of smooth functions $\tilde{U} = (\tilde{u}, \tilde{v})$ and $\hat{U} = (\hat{u}, \hat{v})$ as ordered upper and lower solutions of (1.5) (defined in Definition 1.1), then the reaction-diffusion system (1.5) has a unique solution $U = (u, v)$ with $(\tilde{u}, \tilde{v}) \geq (u, v) \geq (\hat{u}, \hat{v})$ on $(0, \infty)\times \bar{\Omega}$.

It can be easily verified that for any constants $M_1$ and $M_2$ satisfying the relations

the constant functions $(M_1, M_2)$ and $(0, 0)$ are a pair of ordered upper and lower solutions of (1.5) on $(0, \infty) \times \overline{\Omega}$ under any combination of the boundary conditions given in (1.6).

Theorem 1.3. Global existence and boundedness.

For any smooth function $u_0(x)$, $v_0(x)$ on $\overline{\Omega}$ the reaction-diffusion system (1.5) has a unique solution $U = (u, v)$ with $(M_1, M_2) \geq (u, v) \geq (0, 0)$ on $(0, \infty)\times \bar{\Omega}$, with constants $M_1$ and $M_2$ given in (1.9).

2.   Self-elimination by Allee effect

Throughout this paper, for each type of the boundary conditions in (1.6), say $B[\cdot]$, We let $\lambda _0$ and $\phi _0(x)$ (with $\|\phi_0\|_{\infty} = 1$) be the principal eigenvalue and associated eigenfunction of the eigenvalue problem

It is well-known that for Neumann Boundary condition, $\phi _0(x) = 1$ on $\bar{\Omega}$ with $\lambda _0 = 0$. Also, for Dirichlet or Robin boundary condition we have $\phi _0(x) > 0$ in $\Omega$ with $\lambda _0 > 0$. We demonstrate how the Allee effect, birth and death rates, as well as diffusion rates affect the long-term survival of each species with relatively small initial population size.

Theorem 2.1. Self-elimination by Allee effect (Neumann boundary condition).

Under the Neumann boundary conditions for the $u$-species

if $0 \leq u_0(x) < \frac{k_u d_u}{b_u -d_u}$ then $\lim_{t\to\infty} u(t, x) = 0$ uniformly on $\bar{\Omega}$.

Proof. For some $\sigma > 0$, let $\tilde{u} = M e^{-\sigma t}$ and $\hat{u} = 0$, $\tilde{v} = (b_v - d_v)/b_v$ and $\hat{v} = 0$. Since all the defined upper and lower solutions are independent of x, they satisfy the boundary condition inequalities in (1.8) with nonnegative function values and normal derivatives as $0$. Also, $\tilde{u} \geq \hat{u}$ and $\tilde{v} \geq \hat{v}$ on $[0, \infty) \times \bar{\Omega}$. It is obvious that the differential inequalities for $\hat{u}$, $\tilde{v}$ and $\hat{v}$ are satisfied in (1.8).

To verify that $\tilde{u} = M e^{-\sigma t}$ satisfies the corresponding differential inequality in (1.8), we find that the following relation must be satisfied:

After simplifying, the following must hold for all $t \in (0, \infty)$:

It suffices to have

Allowing for a small and positive $\sigma$, for $\tilde{u}$ to function as an upper solution defined in (1.8) we just need:

By Definition 1.1 and Theorem 1.2, for any initial density function $u_0(x)$ with $0 \leq u_0(x) < \frac{k_u d_u}{b_u -d_u}$ on $\overline{\Omega}$, we have

with $M = \|u_0\|_{\infty} < \frac{k_u d_u}{b_u -d_u}$. Therefore, $\lim_{t\to\infty} u(t, x) = 0$ uniformly on $\bar{\Omega}$.

We now analyze the self-elimination by Allee effect of one species (say $u$) under Dirichlet or Robin boundary condition. For this case, the principal eigenvalue for $B_u[\cdot]$ is $\lambda _0 > 0$ and the corresponding eigenfunction $\phi _0 (x) > 0$ in $\Omega$ (with $||\phi _0 ||_{\infty} = 1$).

Theorem 2.2. Self-elimination by Allee effect (Dirichlet or Robin boundary condition).

Under the Dirichlet or Robin (with non-trivial $\gamma_u(x)$) boundary condition for the $u$-species

a) If $D_u \lambda _0 \geq b_u - d_u$, then for any positive $M > 0$ with initial density $u_0(x) \leq M \Phi_0 (x)$, we have $\lim_{t\to\infty} u(t, x) = 0$ uniformly on $\bar{\Omega}$.

b) If $D_u \lambda _0 < b_u - d_u$ and $0 \leq u_0(x) \leq M \Phi_0 (x)$ for some $M < \frac{k_u(\lambda_0 D_u + d_u)}{b_u -d_u-\lambda _0 D_u}$, then $\lim_{t\to\infty} u(t, x) = 0$ uniformly on $\bar{\Omega}$.

Proof. For some positive $\sigma$, Let $\tilde{u} = M \phi _0 e^{-\sigma t}$, and $\hat{u} = 0$, $\tilde{v} = (b_v - d_v)/b_v$ and $\hat{v} = 0$, so $(\tilde{u}, \tilde{v}) \geq (\hat{u}, \hat{v})$ on $[0, \infty) \times \bar{\Omega}$. Since $\phi _0(x) = 0$ on $\partial \Omega$, the boundary inequality in (1.8) for $\tilde{u}$ is satisfied. As seen in the proof of the previous theorem, the differential and boundary inequalities in (1.8) for $\hat{u}$, $\tilde{v}$ and $\hat{v}$ are also satisfied.

For part a), assume that $D_u \lambda _0 > b_u - d_u$. To satisfy the differential inequality in (1.8) for $\tilde{u}$ we need:

Simplifying above and allowing for a small positive $\sigma$, it is suffice to have the following hold for any $t > 0$:

We recognize that $1-M \phi _0 e^{-\sigma t} < 1$ and $0 < \frac{M \phi _0 e^{-\sigma t}}{M \phi _0 e^{-\sigma t}+k_u} < 1$, so we can ensure that:

Hence the differential inequality for the upper solution is satisfied by any $\tilde{u} = M \phi _0 e^{-\sigma t}$ with any $M > 0$. This proves part a).

For part b), assume that $D_u \lambda _0 < b_u - d_u$. As in part a), the differential inequality for $\tilde{u}$ in (1.8 needs to be satisfied for all $t > 0$:

which can be ensured by

The above relation holds for a small and positive $\sigma$, and constant $M$:

By Definition 1.1 and Theorem 1.2, for any initial density function $u_0(x)$ with $0 \leq u_0(x) \leq M \phi_0(x)$ on $\bar{\Omega}$, we have $0 \leq u(t, x) \leq M \phi _0 e^{-\sigma t}$ and $\lim_{t\to\infty} u(t, x) = 0$ uniformly on $\bar{\Omega}$.

Theorems 2.1 and 2.2 shows that larger Allee effect coefficient, higher death rate, and faster diffusion will drive a population to extinction as long as the initial density is within the given ranges for self-elimination.

In the next theorem, we give a result of global extinction under the significance of the Allee effect, for one of the competing species (say $u$) with any initial density size and boundary condition.

Theorem 2.3. Global extinction of $u$-species under Allee effect (any boundary condition).

For the boundary condition $B_u[\cdot] = 0$, let $\lambda _0 \geq 0$ be the principal eigenvalue with the corresponding eigenfunction $\phi _0 (x) > 0$ in $\Omega$. If

then $\lim_{t\to\infty} u(t, x) = 0$ uniformly on $\bar{\Omega}$ with all initial density $0\leq u_0 (x) \leq N \phi _0(x)$ for any $N > 0$.

Proof. Again, for any $N > 0$ and some small $\sigma > 0$, Let $\tilde{u} = N \phi _0 e^{-\sigma t}$, and $\hat{u} = 0$, $\tilde{v} = (b_v - d_v)/b_v$ and $\hat{v} = 0$, so $(\tilde{u}, \tilde{v}) \geq (\hat{u}, \hat{v})$ on $[0, \infty) \times \bar{\Omega}$. We can show see that the differential and boundary inequalities are satisfied are all satisfied for $\hat{u}$, $\tilde{v}$, and $\hat{v}$ under any boundary conditions for $u$ and $v$, as in the proofs for previous theorems.

Also, as seen in (2.6), for the upper solution $\tilde{u}$ to satisfy the required differential inequality given in (1.8), we need to ensure that for all $t > 0$,

We now examine the positive maximum of $G(X) = \frac{b_u (1-X) X}{X + k_u}$ for $X \in [0, \infty)$. Note that $G(0) = G(1) = 0$, and $G(X) < 0$ for $X > 1$.

so $G'(X) = 0$ when $X = -k_u \pm \sqrt{k_u (k_u + 1)}$. We then allocate the only positive maximum point of $G(X)$, at $X = \sqrt{k_u (k_u + 1)}-k_u \in (0, 1)$. This implies that for all $t > 0$,

We can now conclude that if

then for any positive $N$ and some small positive $\sigma$, $0 \leq u(t, x) \leq N \phi _0 e^{-\sigma t}$ on $[0, \infty) \times \bar{\Omega}$ as long as $0 \leq u_0(x) \leq N \phi_0(x)$ on $\bar{\Omega}$. The global extinction of $u$-species given in the theorem then follows.

3.   Permanence

In this section, we explore on conditions for permanence (long-term survival of both species) in the competition model (1.5). Our approach is to find the possibility of a pair of upper-lower solutions given in Definition 1.1 with nontrivial lower solutions for both $u$ and $v$.

Theorem 3.1. Permanence of both species (Neumann boundary condition).

Let $u$ and $v$ both satisfy the no-flux boundary condition $\frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = 0$ on $\partial \Omega$. If

then the competition model (1.5) is permanent as long as the initial density functions satisfy

and

Proof. Let the conditions in (3.1) and (3.2) hold. We will show that $(\tilde{u}, \tilde{v}) = (\frac{b_u-d_u}{b_u}, \frac{b_v-d_v}{b_v})$ and $(\hat{u}, \hat{v}) = (\epsilon _{u}, \epsilon _{v})$, where $\frac{b_u-d_u}{b_u} > \epsilon _u > 0$ and $\frac{b_v-d_v}{b_v} > \epsilon _v > 0$, are a pair of ordered upper and lower solutions given in Definition 1.1. It is clear that $(\tilde{u}, \tilde{v}) \geq (\hat{u}, \hat{v})$. Since $\tilde{u}$, $\hat{u}$, $\tilde{v}$, and $\hat{v}$ are all independent of x, so they satisfy the boundary condition inequalities in (1.8).

In order to satisfy the differential inequalities in (1.8), we need the following relations to hold:

Consider the following factors in the first two inequalities for upper solutions:

and

It's clear to see that the right-hand sides of the first two inequalities are strictly negative, so the differential inequalities for the upper solutions are satisfied.

According to the required differential inequalities for the lower solutions, we need the following to hold:

Multiplying the inequalities in (3.4) by $-\frac{\epsilon _u + k_u}{b_u}$ and $-\frac{\epsilon _v + k_v}{b_v}$ respectively, we now need to analyze the following functions to find $\epsilon _u > 0$ and $\epsilon _v > 0$ to ensure

The functions $F_1$ and $F_2$ (as parabolas opening upward and with $F_1(0)$, $F_2(0) > 0$) have their respective vertices at:

We then see that the following conditions on competition coefficients $\alpha$ and $\beta$ ensure that $\epsilon^*_u$ and $\epsilon^*_v > 0$ in (3.6):

Finally, we place a constraint on $F_1$ and $F_2$ for the function values at the vertices to be negative. For

and doing the same computation for $F_2(\epsilon^*_v) < 0$, we can now find the permanence conditions with respect to the Allee effect coefficients $k_u$ and $k_v$:

Given constraints (3.7) and (3.8), by Definition 1.1 and Lemma 1.2, for any initial density functions $u_0(x)$ and $v_0(x)$ with $\epsilon^*_u \leq u_0(x) \leq \frac{b_u-d_u}{b_u}$ and $\epsilon^*_v \leq v_0(x) \leq \frac{b_v-d_v}{b_v}$, the unique solution of (1.5) exists in $\langle (\hat{u}, \hat{v}), (\tilde{u}, \tilde{v}) \rangle$. This implies permanence in the ecological system and proves the theorem.

In model (1.5), to secure long-term survival of both species we have obtained constraints (3.1) on $\alpha$, $\beta$ (the Lotka-Volterra competition coefficients from one species to another) and $k_u$, $k_v$ (the Allee effect coefficients for each species) in relation with the birth and death rates. Also, the initial density functions need to balanced in ranges given in (3.3). To demonstrate a numerical example (in Figure 1), we make the following choices on the biological parameters which satisfy all conditions for permanence in (3.1):

Also, we set the initial density functions as follows to satisfy the permanence conditions (3.3) on $\overline{\Omega} = [0, 1]$ and the no-flux boundary condition on $\partial \Omega$:

4.   Competitive exclusion

In previous sections, we have studied criteria for self-elimination of each species and long-term survival of both species. Now in this section, we examine conditions for competitive exclusion, that is, one species survives and forces the other species into extinction in (1.5). Various competitive exclusion results can be derived from different combinations of boundary conditions for the two species.

Theorem 4.1. Competitive exclusion of one species ($u$) (Neumann boundary condition).

Let $u$ and $v$ both satisfy the no-flux boundary condition $\frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = 0$ on $\partial \Omega$.Assume that the following inequalities are satisfied

If

then

then

uniformly on $\bar{\Omega}$.

Proof. Let inequalities in (4.1) be satisfied. We have

where $M_1$ is the maximum value of $u_0(x)$ on $\bar{\Omega}$. We are going to verify that for some $\sigma_1 > 0$ (to be determined), $(\tilde{u}, \tilde{v}) = (M_1 e^{-\sigma _1 t}, \frac{b_v-d_v}{b_v})$, and $(\hat{u}, \hat{v}) = (0, \epsilon^*_v)$ are a pair of ordered upper-lower solutions. It is obvious that $(\tilde{u}, \tilde{v}) \geq (\hat{u}, \hat{v})$ in $[0, \infty) \times \bar{\Omega}$. $\tilde{u}$, $\hat{u}$, $\tilde{v}$, and $\hat{v}$ are all independent of x, so their normal derivatives are equal to $0$ on $\partial \Omega$ and the boundary inequalities in (1.8) are satisfied.

In order to satisfy the differential inequalities in (1.8), first notice that the differential inequality for $\hat{u} = 0$ trivially holds. For the rest, we need the following relations to hold:

We now analyze the differential inequalities in (4.2). The second of these inequalities is clearly satisfied by the fact that

Also, by the fact that $M_1 e^{-\sigma _1 t} \leq \frac{b_u-d_u}{b_u}$ and the argument related to $\epsilon^*_v$ in the proof of Theorem 3, we can conclude that the condition $k_v \leq \frac{b_v}{4 d_v}\Big(\frac{b_v -d_v}{b_v}-\beta \frac{b_u-d_u}{b_u}\Big)^2$ ensures the third inequality in (4.2).

We further analyze the first inequality in (4.2). By the fact that $0 \leq e^{-\sigma _1 t} \leq 1$, it suffices to have the following holds and allow a small $\sigma_1 > 0$:

Solving for $M$ from above, we conclude that for $M_1 < \frac{k_u d_u }{b_u -d_u- \alpha b_u \epsilon^*_v}$ and a positive number $\sigma_1$ which is small enough, the first inequality in (4.2) will also be satisfied. Hence we have verified all required inequalities for the coupled upper-lower solutions.

By Definition 1.1 and Lemma 1.2, for any smooth initial density function $u_0(x)$ and $v_0(x)$ with $u_0(x) < M_1$ and $v_0(x) \geq \epsilon^*_v$, there is a unique solution of (1.5) with

This implies that: $\lim_{t\to\infty} u(t, x) = 0$ uniformly on $\bar{\Omega}$ and $v(t, x) \geq \frac{1}{2}\big(\frac{b_v -d_v}{b_v}-\beta \frac{b_u-d_u}{b_u}\big)$ in $[0, \infty) \times \bar{\Omega}$.

To demonstrate a numerical example (in Figure 2) for competitive exclusion of $u$-species given above, we make the following choices on the biological parameters which satisfy all conditions given in (4.1):

Also, we set the initial density functions as follows to satisfy the conditions given in Theorem 4.1 on $\overline{\Omega} = [0, 1]$ and the no-flux boundary condition on $\partial \Omega$:

As seen in the theorem above, from the Allee coefficient $k_u$ and death rate $b_u$ for $u$-species and the persistence strength $\epsilon^*_v$ for $v$-species, we can determine the magnitude $M_1$ of $\|u_0(\cdot)\|_{\infty}$ which will drive the $u$ population to extinction. Moreover, with the presence of a competitor, large competition coefficient ($\alpha$) and small difference between the birth and death rates (ratio $\frac{d_u}{b_u}$ being close to $1$) also will drive the population ($u$) with larger initial size to extinction. The following theorem shows those detrimental effects to the survival of $u$-species under reasonably large initial population size. We now examine conditions for competitive exclusion of $u$-species globally for $0 \leq u_0(x) \leq \frac{b_u-d_u}{b_u}$.

Theorem 4.2. Global exclusion of one species ($u$) under stronger competitor ($v$) (Neumann boundary condition).

Let $u$ and $v$ both satisfy the no-flux boundary condition $\frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = 0$ on $\partial \Omega$. Assume that the inequalities in (4.1) hold. If

with $\epsilon^*_v = \frac{1}{2}\big(\frac{b_v -d_v}{b_v}-\beta \frac{b_u-d_u}{b_u}\big)$, and the initial population functions satisfy $v_0(x) \geq \epsilon^*_v$ and $u_0(x) < \frac{b_u-d_u}{b_u}$, then $\lim_{t\to\infty} u(t, x) = 0$ and $\liminf_{t \to \infty} v(t, x)\geq \frac{1}{2}\big(\frac{b_v -d_v}{b_v}-\beta \frac{b_u-d_u}{b_u}\big)$ uniformly on $\bar{\Omega}$.

Proof. As in the proof of Theorem 4.1, we will verify that for some small $\sigma_1 > 0$ and the maximum of $u_0(x)$ $M_1 < \frac{b_u-d_u}{b_u}$, $(\tilde{u}, \tilde{v}) = (M_1 e^{-\sigma _1 t}, \frac{b_v-d_v}{b_v})$, and $(\hat{u}, \hat{v}) = (0, \epsilon^*_v)$ are a pair of ordered upper-lower solutions. It is already known that $(\tilde{u}, \tilde{v}) \geq (\hat{u}, \hat{v})$ in $[0, \infty) \times \bar{\Omega}$, and all the boundary inequalities in (1.8) are satisfied. Among the four differential inequalities for upper-lower solution given in (4.2), we also know from the previous proof that the second, third, and fourth inequalities already hold.

We now focus on analyzing the first differential inequality in (4.2). Let $M_1 e^{-\sigma _1 t} = X$, we need to find a condition for

with all $X > 0$ and some $\sigma_1 > 0$. Define the function

and note that $H(0) = H(1-\alpha \epsilon^*_v) = 0$, and $H(X) < 0$ for $X > 1-\alpha \epsilon^*_v$. Since $H(x) > 0$ in the interval $(0, 1-\alpha \epsilon^*_v)$, we look for the maximum value of $H(X)$ there. Setting

we find the local extremum at $\sqrt{k_u ^2 +k_u -\alpha \epsilon^*_v k_u}-k_u$ in the interval $[0, 1-\alpha \epsilon^*_v]$.

Since

allowing for a small $\sigma _1 > 0$, the first differential inequality (4.2) is satisfied if

By Definition 1.1 and Lemma 1.2, for any smooth initial density function $u_0(x)$ and $v_0(x)$ with $u_0(x) < \frac{b_u-d_u}{b_u}$ and $v_0(x) \geq \epsilon^*_v$, there is a unique solution of (1.5) with $(0, \epsilon^*_v) \leq (u(t, x), v(t, x)) \leq (M_1 e^{-\sigma _1 t}, \frac{b_v-d_v}{b_v})$. This implies that: $\lim_{t\to\infty} u(t, x) = 0$ uniformly on $\bar{\Omega}$ and $v(t, x) \geq \frac{1}{2}\big(\frac{b_v -d_v}{b_v}-\beta \frac{b_u-d_u}{b_u}\big)$ on $[0, \infty) \times \bar{\Omega}$.

We now examine an numerical example (in Figure 3) for the above global exclusion result, which shows competitive exclusion of $u$-species with any initial density $u_0(x) < (b_u-d_u)/b_u$. We choose the following biological parameters that satisfy all the conditions in (4.1) and (4.3). Also, we use initial density functions with relatively larger $u_0(x)$ which satisfy the conditions given in Theorem 4.2 on $\overline{\Omega} = [0, 1]$ and the no-flux boundary condition on $\partial \Omega$.

At last, we consider the case that the population density function $u(t, x)$ is subject to the Dirichlet or Robin boundary condition (with $\lambda_0 > 0$) and the population density function $v(t, x)$ is subject to the Neumann boundary condition. This way, the $u$-species will be under greater strain at the boundary, so we study the conditions that will drive $u$-species to extinction under Allee effect and competition from $v$-species.

Theorem 4.3. Competitive exclusion of one species (mixed boundary conditions).

Let $u$ and $v$ satisfy the boundary condition

with nontrivial $\gamma_u(x)$ on $\partial \Omega$.

Assume that the inequalities in (4.1) hold and $v_0(x) \geq \epsilon^*_v = \frac{1}{2}\big(\frac{b_v -d_v}{b_v}-\beta \frac{b_u-d_u}{b_u}\big)$, $u_0(x) \leq M \phi_0(x)$ on $\bar{\Omega}$ for some $M > 0$, with eigenfunction $\phi_0(x)$ and eigenvalue $\lambda_0$ for $B_u[\cdot]$ given in (2.1). If one of the following conditions holds:

a)

and $M \leq \frac{b_u-d_u}{b_u}$, or

b)

and $M < \frac{k_u(D_u \lambda_0+d_u}{b_u (1- \alpha \epsilon^*_v)- D_u \lambda _0 -d_u}$, or

c)

and $M \leq \frac{b_u-d_u}{b_u}$, then $\lim_{t\to\infty} u(t, x) = 0$ uniformly on $\bar{\Omega}$ and $v(t, x) > \frac{1}{2}\big(\frac{b_v -d_v}{b_v}-\beta \frac{b_u-d_u}{b_u} \big)$ in $[0, \infty)\times\bar{\Omega}$.

Proof. Once again, for $M$ and $\sigma > 0$, we set up the ordered upper and lower solutions $(\tilde{u}, \tilde{v}) = (M \phi _0 e^{-\sigma t}, \frac{b_v-d_v}{b_v})$, and $(\hat{u}, \hat{v}) = (0, \epsilon*_v)$. Noting that $\tilde{v}$ and $\hat{v}$ are independent of x, with respective normal derivatives $= 0$ on $\partial \Omega$. $\hat{u} = 0$ on $\bar{\Omega}$ and $B_u[\phi _0] = 0$ on $\partial \Omega$. Therefore all boundary inequalities in (1.8) are satisfied.

As seen in the proofs of previous theorems, the differential inequalities in (1.8) for $\tilde{v}$, $\hat{u}$ and $\hat{v}$ also hold as long as $M \leq \frac{b_u-d_u}{b_u}$.

We now analyze the differential inequality (1.8) for $\tilde{u} = M \phi _0 e^{-\sigma t}$:

To satisfy (4.5), it suffices to have the following relation hold which allows a small $\sigma > 0$:

If $b_u (1- \alpha \epsilon^*_v)\leq D_u \lambda _0 +d_u$, the first differential inequality (4.5) is satisfied for all $M > 0$. Therefore the condition a) will ensure that $(\tilde{u}, \tilde{v}) = (M \phi _0 e^{-\sigma t}, \frac{b_v-d_v}{b_v})$ and $(\hat{u}, \hat{v}) = (0, \epsilon*_v)$ are a pair of ordered upper-lower solutions under the mixed boundary conditions.

On the other hand, if $b_u (1- \alpha \epsilon^*_v) > D_u \lambda _0 +d_u$, solving for $M$ from the inequality (4.6) leads to $M < \frac{k_u(D_u \lambda_0+d_u}{b_u (1- \alpha \epsilon^*_v)- D_u \lambda _0 -d_u}$. Hence the condition b) also ensures the ordered upper-lower solutions as designed.

Furthermore, the first differential inequality (4.5) is satisfied if the following holds for any $X > 0$:

Maximizing the right side of the inequality as in the proof of Theorem 4.2, we can conclude that condition c) and a small $\sigma > 0$ ensures the first differential inequality in (4.5) and verifies the ordered upper-lower solutions as designed.

By Definition 1.1 and Lemma 1.2, under the assumptions given and with one condition from a), b), or c), there exists a unique solution of (1.5) with $(0, \epsilon^*_v) \leq (u(t, x), v(t, x)) \leq (M \phi_0 e^{-\sigma t}, \frac{b_v-d_v}{b_v})$. This concludes that: $\lim_{t\to\infty} u(t, x) = 0$ uniformly on $\bar{\Omega}$ and $v(t, x) \geq \frac{1}{2}\big(\frac{b_v -d_v}{b_v}-\beta \frac{b_u-d_u}{b_u}\big)$ in $[0, \infty) \times \bar{\Omega}$.

We will end this section with two numerical examples for Theorem 4.3 on competitive exclusion of one species ($u$) which is under Dirichlet or Robin boundary condition. Setting $\Omega = (0, 1)$, we let $u(t, x)$ be under the Dirichlet boundary condition $u(t, 0) = u(t, 1) = 0$. This way, the principal eigenvalue $\lambda_0 = \pi^2$ with corresponding eigenfunction $\phi_0(x) = \sin(\pi x)$. For condition b) to be satisfied, we choose the following biological parameters and initial functions to demonstrate our numerical simulation (in Figure 4):

For condition c) to be satisfied, we choose the following biological parameters and initial functions to demonstrate our numerical simulation (in Figure 5):

5.   Conclusions and discussion

The results shown in this article demonstrate how the all the biological parameters in the model, especially the coefficients representing Allee effects, ultimately determine the long-term survival of each competing species. We also pay special attention to the impacts of the dimensions of the habitat $\Omega$ and boundary conditions for both species on $\partial \Omega$. As we know, Dirichlet boundary condition ($u = 0$ on $\partial \Omega$) indicates the biological species diminishes on the boundary of the habitat, while Neumann boundary condition ($\frac{\partial u}{\partial \nu} = 0$ on $\partial \Omega$) indicates no flux on the boundary of the habitat as the advection and diffusion fluxes of the population are exactly balanced. And, the Robin boundary condition in (1.6) is a linear combination of those two.

In Section 2, we give several results on self-elimination by Allee effect of one species despite the behavior of other. In Theorem 2.1 (Neumann boundary condition) and Theorem 2.2 (Dirichlet or Robin boundary condition), we find ranges for initial density function $u_0$

which ensure extinction of species $u$. And, in Theorem 2.3, the following self-elimination condition on all the biological parameters is obtained under all boundary conditions and any feasible initial density (global elimination):

We note that

Therefore, for sufficiently large Allee effect coefficient $k_u$, relatively large death rate and limited birth rate, as well as faster diffusion that makes the global elimination condition hold, a species with any nonnegative initial density will be driven to extinction under all three boundary conditions.

In section 3, we explore the possibility of permanence for both species under Neumann (no flux) boundary conditions. This requires balanced constraints on birth, death, and competition rates, as well as the Allee effect coefficients:

Also, the initial density functions are required to be above some minimum thresholds:

and

In Section $4$, conditions for competitive exclusion are investigated under various boundary conditions. Under Neumann boundary conditions for both species and with the persistence constraints for $v$-species hold, we give the range for initial density $u_0$ ensuring extinction of $u$-species and survival of $v$ species in Theorem 4.1:

In Theorems 4.2 and 4.3, under Neumann boundary condition for $v$-species and any boundary condition for $u$-species, global exclusion of $u$-species are ensured by the following condition for all feasible initial density $u_0$:

Here for the $u$-species, $\lambda_0 = 0$ under Neumann boundary condition and $\lambda_0 > 0$ under Dirichlet or Robin boundary condition. Specifically, with Dirichlet or Robin boundary condition, limited habitat size can also be detrimental to the persistence of the $u$-species. Examining a special case of the eigenvalue problem with Dirichlet boundary condition on a $1$-dimensional habitat $\Omega = (0, L)$:

we see that

This shows that smaller habitat size ($L$) makes much bigger $\lambda_0$ which validates the above competitive exclusion condition while all other biological parameters remaining unchanged. In addition to the self-elimination and competitive exclusion constraints on all biological parameters mentioned in Sections $2$ and $4$, this example illustrates that limited habitat size can also drive a biological population into extinction under Dirichlet or Robin boundary condition.

Acknowledgments

The authors would like to thank the anonymous referees for their valuable suggestions to strengthen the results in this article.

Conflict of interest

The authors declare no conflicts of interest.