Bifurcation structure of nonconstant positive steady states for a diffusive predator-prey model

1.   Introduction

Understanding the mechanisms by which patterns are created in the living system poses one of the most challenging problems in developmental biology ^[1], since the pioneer work of Turing ^[2]. Turing's revolutionary idea was that passive diffusion could interact with the chemical reaction in such a way that even if the reaction by itself has no symmetry-breaking capabilities, diffusion can destabilize the symmetry so that the system with diffusion can have them ^[3,4,5,6].

There has been considerable interest in investigating the pattern formation of population system by taking into account the effect of diffusion ^{[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,26,27,28,29,30,31,32,33]}. These investigations have revealed that spatial inhomogeneities like the inhomogeneous distribution of nutrients as well as interactions on spatial scales like migration play an important role in specializing and stabilizing population levels.

In a recent analytic approach by Shi, Li and Lin ^[17], the following reaction-diffusion predator-prey model is considered:

where $u: = u(x, t)$ and $v: = v(x, t)$ represent the densities of the prey and predator, respectively. The parameter $k > 0$ is the maximum consumption rate, $a$ a saturation constant, $m$ a predator interference parameter ($m < 0$ the case where predators benefit from cofeeding ^[18]), $\delta$ the intrinsic growth rate of prey, and $\beta$ the numbers of prey required to support one predator at equilibrium when $v$ equals to $u/\beta$ ^[34]. $d_1, d_2$ are the diffusion coefficients of $u$ and $v$, respectively.

In ^[17], the authors show that, model (1.1) has a boundary steady state $(1, 0)$ which is unstable and a unique positive steady state

And the authors investigated the qualitative properties, including the global attractor, persistence property under the condition of $m > k$, local and global stability of $E^*$, and established the existence and nonexistence of nonconstant positive steady states of model (1.1). In this paper, we always assume that $m > k$.

And there naturally comes a question: What is the structure of nonconstant positive steady states of model (1.1)?

Thus we will investigate the steady state problem corresponding to model (1.1)

It is our purpose in this paper to make a better description for the structure of the set of nonconstant positive steady states of model (1.1). The rest of this article is organized as follows: In Section 2, we prepare some preliminaries and give the Turing instability in details. In Section 3, we give the local and global bifurcation structure of nonconstant positive steady states.

2.   Turing instability

It's well known that the Turing instability refers to "diffusion driven instability", i.e., the stability of the positive equilibrium $E^* = (u^*, v^*)$ changing from stable for the ordinary differential equations (ODE) dynamics (i.e., $d_1 = d_2 = 0$ in model (1.1)), to unstable, for the partial differential equations (PDE) dynamics (1.1) ^[1,14]. The occurrence of Turing pattern is caused by the existence of nonconstant positive steady states of model (1.1) as a result of diffusion. In this section, we mainly discuss Turing instability.

Before proceeding, we recall the following Neumann eigenvalue problem

It is well known that (2.1) has a sequence of simple eigenvalues and associated eigenfunctions are explicitly given by

where $i = 0, 1, 2, \cdot\cdot\cdot$. Let $Y = C^2((0, \pi))\times C^2((0, \pi))$ be the Hilbert space, and

Let us first recall that the ODE model corresponding to the PDE model (1.1):

Following ^[17], the Jacobian of model (2.2) around $E^{*} = (u^*, v^*)$ is given by

where

The characteristic equation of $J$ is

where

It follows from the assumption $m > k$ that $\mathrm{Tr}(J) < 0$. Thus the positive equilibrium $E^*$ of the ODE model (2.2) is locally stable.

Choosing $d_2$ as the bifurcation parameter, we have the following linearized operator of model (1.2) evaluated at $E^*$:

It is easy to see that the eigenvalues of $\mathcal{L}(d_2)$ are given by those if the following operator $\mathcal{L}_i(d_2)$ (see, e.g., ^[16,17,36]):

whose characteristic equation is

where

For simplicity, we define

If $\theta > 0$ and

then what we define as $i_{0}: = i_0(k, a, m, \delta, \beta)$ is the largest positive integer such that $d_{1} i^2 < a_{11}$ with $i\leq i_{0}.$

Clearly, if (2.8) is satisfied, then $1\leq i_{0} \leq\infty$. In this case, we denote

Therefore we can obtain the local stability of $E^* = (u^*, v^*)$ of model (1.1) as follows:

Theorem 2.1. For model (1.1),

(ⅰ) If $\theta < 0$ holds, then $E^*$ is locally asymptotically stable.

(ⅱ) If $\theta > 0$, we have

(ⅰ-1) if $d_1 < a_{11}$ and $0 < d_2 < \bar{d_2}$ hold, then $E^*$ is locally asymptotically stable.

(ⅱ-2) If $d_{1} \leq a_{11}, d_{2} > \bar{d_{2}}$ hold, then $E^*$ is Turing unstable.

Example 2.2. As an example, we take the parameters in model (1.1) as:

Easy to know there is a unique positive equilibrium $E^* = (u^*, v^*) = (0.16548, 0.22064).$ From Theorem 2.1, we can know that if $d_{2} > \bar{d_{2}} = 0.19$ holds, $E^*$ is Turing unstable, and model (1.1) exhibits Turing pattern. In Figure 1, we show the numerical results of model (1.1) with different values of $d_2$. Figure 1a shows Turing pattern with $d_2 = 0.21 > \bar{d_{2}}$, and one can see that the solutions of (1.1) are not dependent on time $t$ but space $x$. In other words, model (1.1) in this case has nonconstant positive steady states as a result of diffusion. Figure 1b gives the stable behavior of model (1.1) with $d_2 = 0.15 < \bar{d_{2}}$.

3.   Bifurcation analysis

In this section, we will focus on the local and global bifurcation structure of the nonconstant positive steady states for model (1.2).

If $(u, v) = (u(x), v(x))$ is a positive solution to model (1.2), then it is proved in ^[17] and the maximum principle ^[35] that

We translate $(u^*, v^*)$ to the origin by the translation $(\tilde{u}, \tilde{v}) = (u-u^*, v-v^*)$. For convenience, we still denote $\tilde{u}, \tilde{v}$ by $u, v,$ respectively, and then we can obtain the following system

3.1.   Local bifurcation

We study the local structure of nonconstant positive solutions for the new system (3.2). In brief, by regarding $d_{2}$ as bifurcation parameter, we verify the existence of positive solutions bifurcation from $(d_{2}, 0, 0)$. In this section, we assume that $\theta > 0$.

With the help of a priori estimate (3.1), let

and $Y = L^p(0, \pi)\times L^p(0, \pi).$ Define the map $F:(0, \infty)\times X\rightarrow Y$ by

Then the solutions of the boundary problem (3.2) are exactly zero of the map $F(d_{2}, u, v).$ Note that $(0, 0)$ is the unique constant solution of (3.2), then we have $F(d_{2}, 0, 0) = 0$. The $\mathrm{Fr\acute{e}chet}$ derivative of $F(d_{2}, u, v)$ with respect to $(u, v)$ at $(0, 0)$ can be given by

where $a_{11}, a_{12}, a_{21}$ and $a_{22}$ are given in (2.4).

Theorem 3.1. Let

(i) Suppose that $d^j_{2}\neq d^i_{2}$ for any integer $i\neq j.$ Then $(d^j_{2}, 0, 0)$ is a bifurcation point of $F(d_{2}, u, v) = 0$. Moreover, there is a one-parameter family of non-trivial solutions $\Gamma_{j}(s) = (d_{2}(s), u(s), v(s))$ of $F(d_{2}, u, v) = 0$ for $|s|$ sufficiently small, where $d_{2}(s), u(s), v(s)$ are continuous functions, $(u(0), v(0)) = (0, 0), d_{2}(0) = d^j_{2}$ and $u(s) = s\varphi_{j}+o(s), v(s) = sb_{j}\varphi_{j}+o(s), b_{j} = \frac{a_{11}-d_{1}\lambda_{j}}{a_{12}} > 0.$ The zero set of $F(d_{2}, u, v)$ consists of two curves $(d_{2}, 0, 0)$ and $\Gamma_{j}(s)$ in a neighborhood of the bifurcation point $(d^j_{2}, 0, 0).$

(ii) Suppose that there exists a positive integer $i(\neq j)$ such that $d^i_{2} = d^j_{2}\triangleq \hat{d}$. Let

If $\dfrac{1+b^*_{i}}{1+b_{i}b^*_{i}}\neq0, \dfrac{1+b^*_{j}}{1+b_{j}b^*_{j}}\neq0$ and $j = 2i (\mathrm{resp}. \, i = 2j)$, then $(\hat{d}, 0, 0)$ is a bifurcation point of $F(d_{2}, u, v) = 0.$ Moreover, there exists a curve of nonconstant solutions $(d_{2}(\omega), s(\omega)(\cos\omega\Phi_{i}+\sin\omega\Phi_{j})+W(\omega))$ of $F(d_{2}, u, v) = 0$ for $\mid\omega-\omega_{0}\mid$ sufficiently small, where $d_{2}(\omega), s(\omega),$ and $W(\omega)$ are continuously differentiable functions with respect to $\omega, W(\omega)\in X_{2}$ and satisfy $d_{2}(\omega_{0}) = \hat{d}, s(\omega_{0}) = 0, W(\omega_{0}) = 0$. Here $\omega_{0}$ is any constant satisfying

where

and

Proof. (ⅰ) By using the Crandall-Rabinowitz bifurcation theorem ^[37], we know that $(d^j_{2};0, 0)$ is a bifurcation point provided that:

(a) the partial derivatives $F_{d_{2}}, F_{(u, v)},$ and $F_{d_{2}, (u, v)}$ exist and are continuous;

(b) $\ker F_{(u, v)}(d^j_{2}, 0, 0)$ and $\mathrm{codim}\, {\mathrm{Im}}\, (F_{(u, v)}(d^j_{2}, 0, 0))$ are one-dimensional (here $\mathrm{Im}$: image);

(c) let $\ker F_{(u, v)}(d^j_{2}, 0, 0) = \mathrm{span}\{\Phi_{j}\},$ then $F_{d_{2}(u, v)}(d^j_{2}, 0, 0)\Phi_{j}\notin \mathrm{Im}(F_{(u, v)}(d^j_{2}, 0, 0)).$

It suffices to verify conditions (a)-(c) above. When $d_{2} = d^j_{2},$ the operator $L_{1}(d_{2})$ is given by

It is clear that the linear operators $F_{(u, v)}, F_{d_{2}, (u, v)}$ and $F_{d_{2}}$ are continuous. The condition (a) is verified.

Suppose $\Phi_{i} = (\bar{\phi}, \bar{\psi})^\top\in\ker L_{1},$ and write $\bar{\phi} = \Sigma \bar{a}_{i}\varphi_{i}, \bar{\psi} = \Sigma \bar{b}_{i}\varphi_{i}.$ Then

By a simple calculation, we have $\det \bar{B}_{i}\neq0$, when $i\neq j$. Hence if and only if $i = j$,

taking $d_{2} = d^j_{2}$ implies that

Consider the adjoint operator

In the same way as above we obtain

Since $\mathrm{Im}(L_{1}) = \ker (L^*_{1})^\bot$ (here $^\bot$: complementary set), thus

Condition (b) is also verified.

Finally, since

and

We can see that $F_{d_{2}(u, v)}(d^j_{2}, 0, 0)\Phi_{j}\notin \mathrm{Im}(L_{1}),$ and so condition (c) is satisfied. The proof of (i) is completed.

(ⅱ) Suppose that there exists a positive integer $i(\neq j)$ such that $d^i_{2} = d^j_{2}\triangleq \hat{d}$. Then

and

which leads to $\mathrm{codim}\; \mathrm{Im}(L_{1}(\hat{d})) = \dim\ker L_{1}(\hat{d}) = 2.$

Clearly, the Crandrall-Rabinowitz bifurcation theorem does not work in the situation since condition (b) in (ⅰ) is not satisfied. Now, we deal with this situation by the techniques of space decomposition and implicit function theorem.

To achieve our aim, we first make the following decomposition

where $X_{1} = \mathrm{span} \{\Phi_{i}, \Phi_{j}\}$ and $X_{2}$ is defined in (3.5). We next look for the solution of $F(d_{2}, u, v)$ in the following form

where $s, \omega\in \mathbb{R}$ are parameters. Define an operator $P$ on $Y$ by

Then $\mathrm{Im}(P) = \mathrm{span}\{\Phi_{i}, \Phi_{j}\} = X_{1}\subset Y, P^2 = P.$ Hence, $P$ is the projection from $Y$ to $X_{1}\subset Y$. Thus we decompose $Y$ as $Y = Y_{1}\oplus Y_{2}$ with $Y_{1}: = \mathrm{Im}(P)$ and $Y_{2}: = \ker P = \mathrm{Im}(L_{1}(\hat{d})).$

Next, we rewrite the map $F:(0, \infty)\times X\rightarrow Y$ by

where

It is obvious that $F(\hat{d}, 0, 0) = 0$, and $F_{(u, v)}(\hat{d}, 0, 0) = L_{1}(\hat{d})$. In order to verify the existence of nonconstant positive solutions of (3.2), we only need to find the existence of nonconstant pair $(u, v)$ satisfying $F(d_{2}, u, v) = 0$.

Fixing $\omega_{0}\in \mathbb{R}$ for the time being, we define a nonlinear mapping

by

where

It is clear that $K(\hat{d}, 0, 0;\omega_{0}) = 0.$ By some calculations, we see that the Frechet derivative of $K(d_{2}, s, W; \omega)$ with respect to $(d_{2}, s, W)$ at $(d_{2}, s, W; \omega) = (\hat{d}, 0, 0;\omega_{0})$ is the liner mapping

where $A_{k}$ and $B_{k}$ ($k = 1, 2, 3$) are given in (3.8).

We then show that

is an isomorphism. To this end, we rewrite

where $\Upsilon_{1}\in Y_{1}, \Upsilon_{2}\in Y_{2}$, and we decompose

where

Furthermore, we can easily check that

In the next moment, we shall divide our discussion into two cases $j = 2i$ and $i = 2j$.

Case 1: $j = 2i$. In this case, a simple calculation yields

Then, it is clear that

We decompose

where

By the decomposition of $Y$, we have

Let $K_{(d_{2}, s, W)}(\hat{d}, 0, 0;\omega_{0})(d_{2}, s, W) = 0,$ then we get $\Upsilon_{1} = 0$ and $\Upsilon_{2} = 0$. Since $\omega_{0}$ satisfies (3.6) when $j = 2i$, we obtain $d_{2} = 0$ and $s = 0$ from $\Upsilon_{1} = 0$. Embedding them into $\Upsilon_{2} = 0$, we have $W = 0$. This shows that $K_{(d_{2}, s, W)}(\hat{d}, 0, 0;\omega_{0})$ is injective.

We now prove that $K_{(d_{2}, s, W)}(\hat{d}, 0, 0;\omega_{0})$ is surjective. For any $(u, v)\in Y$, we need to find $(d_{2}, s, W)\in \mathbb{R}^+\times \mathbb{R}\times X_{2}$ such that

By the decomposition of $Y$, there exist $\alpha, \beta\in \mathbb{R}$ and $(\bar{u}, \bar{v})\in Y_{2}$ such that

Substituting it into (3.10), we obtain

By (3.6), we obtain

Note that $L_{1}(\hat{d})$ is an isomorphism from $X_{2}$ to $Y_{2}$, and we get

by embedding $d_{2} = \tilde{d_{2}}$ and $s = \tilde{s}$ into the third equation of (3.11), where

Then

is the solution of (3.10), which implies $K_{(d_{2}, s, W)}(\hat{d}, 0, 0;\omega_{0})$ is surjective.

Applying the implicit function theorem to

we can know that there is a curve of nonconstant solutions $(d_{2}(\omega), s(\omega), W(\omega))$ of (3.12) in small neighborhood of $\omega_{0}$, where $\omega_{0}$ satisfies (3.6), $d_{2}(\omega), s(\omega)$ and $W(\omega)$ are continuously differentiable functions and satisfy $d_{2}(\omega_{0}) = \hat{d}, s(\omega_{0}) = 0, W(\omega_{0}) = 0.$ Therefore, $(d_{2}(\omega), s(\omega)(\cos\omega\Phi_{i}+\sin\omega\Phi_{j})+W(\omega))$ are nonconstant solutions of $F(d_{2}, (u, v)) = 0$.

Case 2: $i = 2j$. A simple calculation yields

Then

We decompose

where

Hence, we have

As in Case 1 above, if $\omega_{0}$ satisfies (3.7), then $K_{(d_{2}, s, W)}(\hat{d}, 0, 0;\omega_{0})$ is an isomorphism from $:\mathbb{R}^+\times \mathbb{R}\times X_{2}$ to $Y$. By the implicit function theorem, we finish the proof of this case. Thus the whole proof is completed.

Remark 3.2. In Theorem 3.1 (ii), the existence of nonconstant positive solution of (3.2) is showed. Indeed, we derive many curves of nonconstant solutions because $\omega_{0}$ is not unique and can be chosen only to satisfy (3.6) or (3.7). Moreover, due to $b_{i}\neq b_{j}$, it is impossible that both $A_{k}$ and $B_{k}, k = 1, 2, 3,$ are simultaneously equal to zero, which implies that there must exists $\omega_{0}$ satisfying the condition (3.6) above. Similarly, we can check that there must exists $\omega_{0}$ satisfying the condition (3.7) above.

3.2.   Global bifurcation

Theorem 3.1 provides no information of the bifurcation curve $\Gamma_{j}$ far from the equilibrium. In order to understand its global structure, a further study is therefore necessary.

We first introduce the standard abstract bifurcation theorem from ^[37] for readers' convenience. Let $X$ be a Banach space and let $T :\mathbb{R} \times X \rightarrow X$ be a compact, continuously differentiable operator such that $T (a, 0) = 0$. Assume that $T$ can be written as

where $K(a)$ is a linear compact operator and the Fréchet derivative $W_U (a, 0) = 0$. Regarding $a$ as a bifurcation parameter, we will undertake a global bifurcation analysis for the equation

We suppose that $\mathbb{I}- K : X\rightarrow X$ is a bijection. Then the Leray-Schauder degree

where $\hat{B}$ is a ball centered at $0$ in $X$ and $p$ is the sum of the algebraic multiplicities of the eigenvalues of $K$ that are larger than $1$. If $x_0$ is an isolated fixed point of the operator $T$ and $B$ is a ball centered at $x_0$ such that $x_0$ is the unique fixed point of $T$ in $B$, the index of $T$ at $x_0$ is defined as

Moreover, if $x_0$ is a fixed point of $T$ and $I-T^*(x_0)$ is invertible, then $x_0$ is an isolated fixed point of $T$ and

where $B$ and $\hat{B}$ are sufficiently small.

We now state the result on the global bifurcation for the operator $T$ defined by (3.14).

Lemma 3.3. [37, Theorem 1.3] Let $a_0$ be such that $\mathbb{I}-K(a)$ is invertible if $0 < |a-a_0| < \epsilon$ for $\epsilon > 0$. Assume that $\mathrm{index}\, (T (a, \cdot), 0)$ is constant on $(a_0- \epsilon, a_0)$ and on $(a_0, a_0+ \epsilon)$; moreover, if $a_0-\epsilon < a_1 < a_0 < a_2 < a_0 +\epsilon$, then $\mathrm{index}(T (a_1, \cdot), 0) = \mathrm{index}(T (a_2, \cdot), 0).$ Then there exists a continuum $C$ in the $a-U$ plane of solutions of (3.14) such that one of the following alternatives is true

(i) $C$ joins $(a_0, 0)$ to $(\hat{a}, 0)$ where $\mathbb{I}-K(\hat{a})$ is not invertible;

(ii) $C$ joins $(a_0, 0)$ to $\infty$ in $\mathbb{R}\times X$.

Theorem 3.4. Under the same assumptions as in Theorem 3.1, the projection of the bifurcation curve $\Gamma_{j}$ is on the $d_{2}$-axis contains $(d^j_{2}, \infty).$ If $d_{2} > \bar{d_{2}}$ and $d_{2}\neq d^k_{2}$ for any integer $k > 0,$ then model (1.2) possesses at least one non-constant positive solution, where $\bar{d_{2}} = \min_{0\leq i \leq i_{0} }d_{2}^i$.

Proof. First, we rewrite model (1.2) in a form that the standard global bifurcation theory can be more conveniently applied.

Let $\tilde{u} = u-u^*, \tilde{v} = v-v^*,$ then (1.2) is transformed into

where $h_{1}(\tilde{u}, \tilde{v}), h_{2}(\tilde{u}, \tilde{v})$ are higher-order terms of $\tilde{u}$ and $\tilde{v}.$ The constant steady state $(u^*, v^*)$ of (1.2) shifts to $(0, 0)$ of this new system.

Let $G_{1}: h\rightarrow\omega$ denote the Green operator for the boundary value problem

and $G_{2}: h\rightarrow\omega$ the Green operator for

where $a_{11} > 0$ and $a_{22} < 0$. Put $\tilde{U} = (\tilde{u}, \tilde{v}),$

and

Recall that

Then the boundary value problem (3.15) can be interpreted as the equation

Note that $K(d_{2})$ is a compact liner operator on $X$ for any given $d_{2} > 0, H(\tilde{U}) = o(|\tilde{U}|)$ for $\tilde{U}$ near zero uniformly on closed $d_{2}$ sub-intervals of $(0, \infty)$, and is a compact operator on $X$ as well.

In order to apply Rabinowitz's global bifurcation theorem, we first verify that 1 is an eigenvalue of $K(d^j_{2})$ of algebraic multiplicity one. From the argument in the proof of Theorem 3.1 it is seen that $\ker (K(d^j_{2})-I)) = \ker L_{1} = \mathrm{span}\{\Phi_{j}\},$ so 1 is indeed an eigenvalue of $K = K(d^j_{2}),$ and $\dim \ker(K-I) = 1$. As the algebraic multiplicity of the eigenvalue 1 is the dimension of the generalized hull space $\cup_{i = 1}^{\infty}\ker(K-I)^i,$ we need to verify that $\ker(K-I) = \ker(K-I)^2,$ or $\ker(K-I)\cap R(K-I) = {0}.$

We now compute $\ker(K^*-I)$ following the calculation in ^[36], where $K^*$ is the adjoint of $K$.

Let $(\hat{\phi}, \hat{\psi})\in\ker(K^*-I),$ then

By the definition of $G_{1}$ and $G_{2}$ we obtain

where

Let $\hat{\phi} = \sum_{i = 0}^{\infty}\hat{a}_{i}\varphi_{i}, \hat{\psi} = \sum_{i = 0}^{\infty} \hat{b}_{i}\varphi_{i},$ then

By a straightforward calculation one can check that $\det\hat{B}_{i} = a_{12}\det\bar{B}_{i},$ where $\bar{B}_{i}$ is given in (3.9) (replacing $d_{2}$ there by $d^j_{2}).$ Thus $\det\bar{B}_{i} = 0$ only for $i = j$, and $\ker(K^*-I) = \mathrm{span}\{\hat{\Phi}_{j}\}$, where $\hat{\Phi}_{j} = (d_{1}\lambda_{i}+a_{11}, 1)^\top\varphi_{j}.$ This shows that $\Phi_{j}\notin(\ker(K^*-I))^\bot = R(K-I),$ so $\ker(K-I)\cap R(K-I) = {0}$ and the eigenvalue 1 has algebraic multiplicity one.

If $0 < d_{2}\neq d^j_{2}$ is in a small neighborhood of $d^j_{2},$ then the liner operator $I-K(d_{2}):X\rightarrow X$ is a bijection and 0 is an isolated solution of (3.15) for this fixed $d_{2}$. The index of this isolated zero of $I-K(d_{2})-H$ is given by

where $B$ is a sufficiently small ball center at 0, and $p$ is the sum of the algebraic multiplicities of the eigenvalues of $K(d_{2})$ lager than 1.

For our bifurcation analysis, it is also necessary to show that this index changes as $d_{2}$ crosses $d^j_{2},$ that is, for $\epsilon > 0$ sufficiently small, we need to verify

Indeed, if $\mu$ is an eigenvalue of $K(d_{2})$ with an eigenfunction $(\hat{\phi}, \hat{\psi}),$ then

By using the Fourier cosine series $\hat{\phi} = \Sigma \hat{a}_{i}\varphi_{i}$ and $\hat{\psi} = \Sigma \hat{b}_{i}\varphi_{i}$, we have

Thus the set of eigenvalues of $K(d_{2})$ consists of all $\mu$ which implies the following characteristic equation

where the integer $i$ runs from zero to $\infty$. In particular, for $d_{2} = d^j_{2}$, if $\mu = 1$ is a root of (3.18), then a simple calculation leads to $d^j_{2} = d^i_{2},$ and so $j = i$ by the assumption. Therefore, without counting the eigenvalues corresponding to $i\neq j$ in (3.18), $K(d_{2})$ has the same number of eigenvalues greater than 1 for all $d_{2}$ close to $d^j_{2},$ and they have the same multiplicities. On the other hand, for $i = j$ in (3.18), we let $\mu(d_{2}), \tilde{\mu}(d_{2})$ denote the two roots of (3.18). By a straightforward calculation, we find that

Now for $d_{2}$ close to $d^j_{2},$ we obtain $\tilde{\mu}(d^j_{2}) < 1.$ As the constant term $-a_{12}a_{21}/d_{2}\lambda_{i}-a_{22}$ in (3.18) is a decreasing function of $d_{2}$, there exists

Consequently, $K(d^j_{2}+\epsilon)$ has exactly one more eigenvalues that are larger than 1 than $K(d^j_{2}-\epsilon)$ does, and by a similar argument above we can show this eigenvalue has algebraic multiplicity one. This verifies (3.17).

Therefore, we apply Theorem 1.3 in ^[37] to conclude that $\Gamma _j$ either

(1) meets infinity in $\mathbb{R} \times \mathrm{X}$, or

(2) meets $(d^k_2, (u^*, v^*))$ for some $k \neq j, d^k_2 > 0$.

We show that $\Gamma _j$ must extend to infinity in $\mathbb{R}\times \mathrm{X}$ by the idea of Nishiura ^[38] and Takagi ^[39]. Thus, the theorem is verified.

Remark 3.5. Theorem 3.4 shows that there is a smooth curve $\Gamma_{j}$ of positive solutions of model (1.2) bifurcating from $(d^j_{2}, u^*, v^*)$, with $\Gamma_{j}$ contained in a global branch of the positive solutions of (1.2).

4.   Discussion and conclusion

In this paper, based on the results in ^[17], we make a detailed descriptions for the local (c.f., Theorem 3.1) and global (c.f., Theorem 3.4) bifurcation structure of nonconstant positive steady states of a modified Holling-Tanner predator-prey system under homogeneous Neumann boundary condition. These theorems and the results in ^[17] can give a profile of the solutions of model (1.1). The results are beneficial to population persistence control, that is, we must do our best to regulate the parameters in the special range to avoid population extinction.

It is should be noted that our results are based on 1-dimensional space. And in the $N$-dimensional space, $N\geq 2$, we can also obtain a local bifurcation result, which is an analogue of Theorem 3.1. But the global bifurcation (Theorem 3.4) is only established in the square domain. For instance, in the special 2-D case with $(0, L) \times(0, L)$. In this special case, the Neumann eigenvalue problem has eigen-pairs

Each eigenvalue gives rise to bifurcation point $d_2^{mn}$, with $\lambda_j$ being replaced by $\lambda_{mn}$ in (3.3). We mention that though the boundary of the square is not smooth, the Neumann boundary condition has to be interpreted in the weak fashion via first Green's identity in the standard way ^[40].

Acknowledgments

This research was supported by the National Science Foundation of China (11601179, 61672013 and 61772017) and Huaian Key Laboratory for Infectious Diseases Control and Prevention (HAP201704), Huaian, Jiangsu Province, China.

Conflict of interest

The authors declare that they have no competing interests.