The dynamics of a delayed predator-prey model with square root functional response and stage structure

1.   Introduction

This paper concerns the existence of $\tau$-periodic brake orbits ($\tau > 0$) of the autonomous first-order Hamiltonian system

where $H\in C^{2}(\mathbb{R}^{2n}, \mathbb{R})$ with $H(Nz) = H(z)$, $z\in \mathbb{R}^{2n}$, $J = \left(\begin{array}{cc} 0 & -I_n\\ I_n & 0\end{array}\right)$ and $N = \left(\begin{array}{cc} -I_n & 0\\ 0 & I_{n}\end{array}\right)$ with $I_{n}$ the $n\times n$ identity matrix.

As shown in ^[1,2], for $\vec{x} = (x_1, \cdots, x_n)$ and $\vec{y} = (y_1, \cdots, y_n)$, we set

For $z = (p_{1}, \cdots, p_{n}, q_{1}, \cdots, q_{n})$, we have

Below are the conditions cited from ^[3] with minor modifications.

(H1) $H\in C^{2}(\mathbb{R}^{2n}, \mathbb{R})$, $H(Nz) = H(z), \ z\in \mathbb{R}^{2n}$.

(H2) There exist $\gamma_{i} > 0$ $(i = 1, \ \cdots, \ n)$ such that

where $\omega(z) = \sum\limits_{i = 1}^{n}\left(| p_{i}|^{1+\gamma_{i}} +| q_{i}|^{1+\frac{1}{\gamma_{i}}}\right)$.

(H3) There exist $\beta > 1$ and $c_{1}, \ c_{2}, \ \alpha_{i}, \ \beta_{i} > 0$ with $\alpha_{i}+\beta_{i} = 1\ (1\leq i\leq n)$ such that

where $V(z) = V(\vec{\alpha}, \vec{\beta})(z)$ with $\vec{\alpha} = (\alpha_1, \cdots, \alpha_n)$, $\vec{\beta} = (\beta_1, \cdots, \beta_n)$.

(H4) There exists $\lambda\in[1, \frac{\beta^2}{\beta+1})$ such that

where $H^{\prime\prime}_{zz}$ means the Hessian matrix of $H$.

(H5) $H(0) = 0$ and $H(z) > 0, \ |\nabla H(z)| > 0$ for $z\neq 0$.

Note that (H2) is a variant subquadratic growth condition which has superquadratic growth behaviors in some components and has subquadratic growth behaviors in other components, while ^[4] provided one other kind of variant subquadratic growth condition, we also call such conditions anisotropic growth conditions.

In the last decades, brake orbit problems have been investigated deeply, see ^{[5,6,7,8,9,10,11,12,13]} and references therein. In ^[14], the existence of brake orbits and symmetric brake orbits were proved under the classical superquadratic growth conditions. Meanwhile, the minimal period estimates were given by comparing the $L_{0}$-index iterations. Later, in ^[15], the authors obtained the same minimal period estimates under a weak growth condition which has super-quadratic growth only on some $J$-invariant plane. In ^[4,16], the authors considered first-order anisotropic convex Hamiltonian systems and reduced the existence problem of brake orbits to the dual variation problem, moreover, in ^[4], the minmality of period for brake orbits was obtained. In ^[1], the authors removed the convex assumption in ^[16] and obtained brake orbits with minimal period estimates under more general anisotropic growth conditions which are variant superquadratic growth conditions.

The following is the main result of this paper.

Theorem 1.1. If $H$ is a Hamiltonian function satisfying (H1)–(H5), then there exists $\tilde{\tau} > 0$ such that when $\tau\geq\tilde{\tau}$, the system (1.1) has a nontrivial brake orbit $z$ with the $L_{0}$-index estimate

Futhermore, if the above brake orbit z also satisfies

(H6) $H^{\prime\prime}_{zz}(z(t))\geq 0$, $t\in \mathbb{R}$ and $\int_{0}^{\frac{\tau}{2}}H_{qq}''(z(t)){\mathrm{d}}t > 0$, where $H_{qq}''(z)$ means the Hessian matrix w.r.t. $q$ for $z = (p, q)$, $p, \ q\in\mathbb{R}^{n}$.

Then the brake orbit z has minimal period $\tau$ or $\frac{\tau}{2}.$

We remind the readers that the minimal period $\frac{\tau}{2}$ may not be eliminated generally. See Remark 4.2 in ^[14], for example, the minimal period is $\frac{\tau}{2}$ under the condition (H6). In ^[2], we also consider the symmetric brake orbit case under the above conditions with small changes using different index iteration inequalities.

If $\tilde{z}$ is a brake orbit for the system (1.1), then $z(t) = \tilde{z}(\frac{\tau}{2} t)$ satisfies

The converse is also true. So finding brake orbits for the system (1.1) is equivalent to finding 2-periodic brake orbits for the system (1.3).

In Section 2, we recall the $L_{0}$-index theory and the related Sobolev space. In Section 3, we prove the existence of a nontrivial brake orbit with minimal period $2$ or $1$.

2.   Preliminaries

The Maslov-type index theory is higly-developed and widly-used to study the existence, minimality of period, multiplicity and stability of periodic solutions of Hamiltonian systems, see ^[17]. And to estimate the minimal period for brake orbits, Liu and his cooperators introduced the $L_{0}$-index theory —a topologically variant Maslov-type index theory, see the monograph ^[18] and the recent survey paper ^[19].

We denote by ${{\cal L}} (\mathbb{R}^{2n})$ the set of all $2n\times2n$ real matrices, and denote by ${{\cal L}}_{s}(\mathbb{R}^{2n})$ its subset of symmetric ones. The symplectic group ${\rm{Sp}}(2n)$ for $n \in \mathbb{N}$ and the symplectic path $\mathcal{P}_{\tau}(2n)$ in ${\rm{Sp}}(2n)$ starting from the identity $I_{2n}$ on $[0, \tau]$ are denoted respectively by

As showed in ^[18], for the Lagrangian subspaces $L_{0} = \{0\}\times \mathbb{R}^{n}$ and $L_{1} = \mathbb{R}^{n}\times\{0\}$, there are two pairs of integers $(i_{L_{k}}(\gamma, \tau), \nu_{L_{k}}(\gamma, \tau)) \in \mathbb{Z}\times \{0, \ 1, \ \cdots, \ n\}$ $(k = 0, 1)$ associated with $\gamma \in \mathcal{P}_{\tau}(2n)$ on the interval $[0, \tau]$, called the Maslov-type index associated with $L_{k}$ for $k = 0, 1$ or the $L_{k}$-index of $\gamma$ in short. When $\tau = 1$, we simply write $(i_{L_{k}}(\gamma), \nu_{L_{k}}(\gamma))$.

The $L_{0}$-iteration paths $\gamma^{j}: [0, j]\rightarrow {\rm{Sp}}(2n)$ of $\gamma\in \mathcal{P}_{1}(2n)$ (see ^[18]) are defined by

and more generally, for $j\in\mathbb{N}$,

Then we denote by $(i_{L_{0}}(\gamma^{j}), \nu_{L_{0}}(\gamma^{j}))$ the $L_{0}$-index of $\gamma^{j}$ on the interval $[0, j]$.

Assume $B(t)\in C([0, \tau], \mathcal{L}_{s}(\mathbb{R}^{2n}))$ satisfies $B(t+\tau) = B(t)$ and $B(\frac{\tau}{2}+t)N = NB(\frac{\tau}{2}-t)$, consider the fundamental solution $\gamma_{B}$ of the following linear Hamiltonian system

Then $\gamma_{B}\in \mathcal{P}_{\tau}(2n)$. Note that $\gamma_{B}^{k}$ satisfies

The $L_{0}$-index of $\gamma_{B}$ is denoted by $(i_{L_{0}}(B), \nu_{L_{0}}(B))$, called the $L_{0}$-index pair with respect to $B.$

Moreover, if $z$ is a brake orbit of the system (1.1), set $B(t) = H''(z(t))$, denote by $(i_{L_{0}}(z), \nu_{L_{0}}(z))$ the $L_{0}$-index of $\gamma_{B}$, called the $L_{0}$-index pair with respect to $z.$

See ^[17] for the Maslov-type index $(i_{1}(\gamma), \nu_{1}(\gamma))$ of $\gamma\in\mathcal{P}(2n)$. And we refer to ^[18] for the indices $(i^{L_{0}}_{\sqrt{-1}}(\gamma), \nu^{L_{0}}_{\sqrt{-1}}(\gamma))$ and $(i^{L_{0}}_{\sqrt{-1}}(B), \nu^{L_{0}}_{\sqrt{-1}}(B))$ for $\tau = 1$.

Below are some basic results needed in this paper.

Lemma 2.1. (^[11]) For $\gamma\in\mathcal{P}(2n)$, there hold

Lemma 2.2. (^[14]) Suppose $B(t)\in C([0, 2], \mathcal{L}_{s}(\mathbb{R}^{2n}))$ with $B(t+2) = B(t)$ and $B(1+t)N = NB(1-t)$. If $B(t)\geq0$ for all $t\in[0, 2]$, then

Lemma 2.3. (^[14]) Suppose $B(t)\in C([0, 2], \mathcal{L}_{s}(\mathbb{R}^{2n}))$ with $B(t+2) = B(t)$ and $B(1+t)N = NB(1-t)$. If $B(t) = \left(\begin{array}{cc} S_{11}(t) & S_{12}(t) \\ S_{21}(t) & S_{22}(t) \\ \end{array} \right)\geq0$ and $\int_{0}^{1}S_{22}(t)\; {\mathrm{d}}t > 0$, then $i_{L_{0}}(B)\geq0$.

Lemma 2.4. (^[18]) The Maslov-type index iteration inequalities are presented below.

$1^\circ$ For $\gamma\in\mathcal{P}(2n)$ and $k\in 2\mathbb{N}-1$, there holds

$2^\circ$ For $\gamma\in\mathcal{P}(2n)$ and $k\in 2\mathbb{N}$, there holds

Now we introduce the Sobolev space $E = W_{L_{0}}$ and its subspaces as in ^[10,14].

For $m\in \mathbb{N}$, define

and set $E_{m}^{+}: = E_{m}\cap E^{+}$, $E_{m}^{-}: = E_{m}\cap E^{-}$. Then $E = E^{0}\oplus E^{-} \oplus E^{+}$ and $E_{m} = E^{0}\oplus E_{m}^{-} \oplus E_{m}^{+}$. Moreover $\{E_{m}, P_{m}\}$ forms a Galerkin approximation scheme of the unbounded self-adjoint operator $-J\frac{{\mathrm{d}}}{{\mathrm{d}}t}$ defined on $L^{2}([0, 2]; L_{0})$, where $P_{m}: E\rightarrow E_{m}$ denotes the orthogonal projection. Furthermore, define the following bounded self-adjoint operator $\mathcal{A}$ on $E$

and, obviously, $\langle \mathcal{A}z, z\rangle = 2(\|z^{+}\|^{2}-\|z^{-}\|^{2}), \ \mathcal{A}z = \mathcal{A}z^{+}-\mathcal{A}z^{-}$, $z\in E.$

Remark 2.1. (^[1]) For $z\in E$, there holds $V(\vec{x}, \vec{y})z\in E$. And for $z\in E_{m}$, we have $V(\vec{x}, \vec{y})z\in E_{m}$. As for the Fourier expression for $V(\vec{x}, \vec{y})z$, see ^[1] for details. Note that for $V$ defined in (H2) and $z\in E$, we have $V(z)\in E$. Moreover, a simple computation shows that

In our case, assume $B(t)\in C([0, 2], \mathcal{L}_{s}(\mathbb{R}^{2n}))$ satisfies $B(t+2) = B(t)$ and $B(1+t)N = NB(1-t),$ define the following bounded self-adjoint compact operator $\mathcal{B}$

For any $d > 0$, denote by $M_{d}^{-}(\cdot), \ M_{d}^{0}(\cdot), \ M_{d}^{+}(\cdot)$ the eigenspaces corresponding to the eigenvalues $\lambda$ belonging to $(-\infty, -d], \ (-d, d), \ [d, +\infty)$ respectively. Set $(\mathcal{A}-\mathcal{B})^{\sharp} = (\mathcal{A}-\mathcal{B}|_{{\rm{Im}}(\mathcal{A}-\mathcal{B})})^{-1}$. The following result is crucial to esmiate the $L_0$-index.

Lemma 2.5. (^[20,21]) For $B(t)\in C([0, 2], \mathcal{L}_{s}(\mathbb{R}^{2n}))$ satisfying $B(t+2) = B(t)$, $B(1+t)N = NB(1-t)$ and $0 < d \leq \frac{1}{4}\left\|(\mathcal{A}-\mathcal{B})^{\sharp}\right\|^{-1}$, there exists $m_{0} > 0$ such that for $m \geq m_{0}$, we have

3.   Main results

As shown in ^[10,14], searching for brake orbits for the system (1.3) can be transformed into finding critical points of the following functional

By (H4), we have $g\in C^{2}(E, \mathbb{R})$, then, let us now set $g_{m} = g|_{E_m}, \ m\in \mathbb{N}$. To find the critical points of $g_{m}$, we shall prove that $g_{m}$ satisfies the hypotheses of the homological link Theorem 4.1.7 in ^[22]. The following several lemmas are essential.

Lemma 3.1. If H(z) satifies $(H1)$, $(H3)$ and $(H4)$, then the above functional $g$ satisfies (PS)$^{\ast}$ condition with respect to $\{E_m\}_{m\in\mathbb{N}}$, i.e., any sequence $\{z_{m}\} \subset E$ satisfying $z_{m} \in E_m$, $g_{m}(z_{m})$ is bounded and $\nabla g_{m}(z_{m}) \to 0$ as $m\to +\infty$ possesses a convergent subsequence in $E$.

Proof. We follow the ideas in ^[3].

Let $\{z_{m}\}$ be a sequence such that $| g(z_{m})|\leq c_{3}$ and $\nabla g_{m}(z_{m}) \to 0$ as $m \to \infty$, where $c_3 > 0$. To prove the lemma, it is enough to show that $\{z_{m}\}$ is bounded.

For $m$ large enough, by Remark 2.1 and (H3), we have

then there exists $c_{4} > 0$ such that

For large $m$, we have

By (3.2), (H4), H$\ddot{{\rm{o}}}$lder's inequality and the embedding theorem, we obtain

where $\beta > \lambda\geq1$ for (H3), (H4) and $c_{5}, \ c_{6} > 0$ are suitable constants.

Combining (3.1) and (3.3), for $m$ large enough, there exists $c_{7} > 0$ such that

Set $\widehat{z}_m = z_m-z^0_m = z^+_m+z_m^-.$ By (H4), (3.4) and the embedding theorem, we obtain

where $c_{8},$ $c_9 > 0$ are suitable constants. From (3.4) and (3.5), we see

where $c_{10} > 0$. From (H3), it follows that

From (3.6) and (3.7), we see that

where $c_{11} > 0$. From (3.4), (3.8) and $\frac{\lambda+\lambda\beta}{\beta^{2}} < 1$, we see $\{z_{m}\}$ is bounded.

For $u_{0}\in E^+_1$ with $\|u_{0}\| = 1$, define $S = \left(E^-\oplus E^0\right)+u_{0}$.

Lemma 3.2. If H(z) satifies $(H1)$, $(H4)$ and $(H5)$, then there exists $\tilde{\tau} > 0$ such that for $\tau\geq\tilde{\tau}$, there holds $\inf\limits_{S} g > 0$.

Proof. The ideas come from ^[23].

For $z\in S$, we have

There exist two cases to be considered.

Case (i) If $\| z^{-}\| > 1$, then by (H5), we have

Case (ii) If $\| z^{-}\|\leq1,$ set $\Omega = \{z\in S \mid \| z^{-}\|\leq1\}$, then $\Omega$ is weakly compact and convex.

Since the functional $z\mapsto\int_{0}^{2}H(z)\; {\mathrm{d}}t$ is weakly continuous, then the functional achieves its minimum on $\Omega$, assume the minimum is $\sigma$ achieved at $u^{-}+u_0\in S$. Since $u_0 \neq 0$, we have $u^{-}+u_0\neq 0$, then $\sigma > 0$ by (H5).

Set $\tilde{\tau} = \frac{2}{\sigma}$, for $\tau > \tilde{\tau}$, by (3.9), we have

Therefore, the lemma holds.

Choose $\mu > 0$ large enough such that $\sigma_{i} = \frac{\mu}{1+\gamma_{i}} > 1$ and $\tau_{i} = \frac{\mu}{1+\frac{1}{\gamma_{i}}} > 1$. For $\rho > 0$, we set

where $z = (p_1, \cdots, p_{n}, q_1, \cdots, q_{n})\in E$. Note that $L_{\rho}$ is well-defined on $E$ by Remark 2.1. The operator $L_{\rho}$ is linear bounded and invertible and $\|L_\rho\|\leq1$, if $\rho\leq1$.

For any $z = z^0+z^{-} +z^{+} \in E$, we have

Lemma 3.3. If $H$ satisfies (H2), then there exists $\rho > 1$ large enough such that $\sup\limits_{L_{\rho}(\partial Q)} g < 0$, where $Q = \{z\in E^{+}\mid \| z\|\leq \rho\}$.

Proof. For any $\epsilon > 0$, by $(H2)$, there exists $M_{\epsilon}$ such that

For $z\in\partial Q$, from (3.10) and (3.11), we have

where $c_{12} > 0$ is the embedding constant.

Choose $\epsilon > 0$ such that $n\tau\epsilon c_{12} < 1$, then for $\rho > 1$ large enough, we have $\sup\limits_{L_{\rho}(\partial Q)} g < 0$.

Lemma 3.4. Set $S_{m} = S\cap E_{m}$ and $Q_{m} = Q\cap E_m$. For $\rho > 1$ defined as above, we have $L_{\rho}(\partial Q_{m})$ and $S_{m}$ homologically link.

Proof. Since $\rho > 1$, $\rho > \| L_{\rho}^{-1}\| = \| L_{\frac{1}{\rho}}\|$. By direct computation, we can check that $PL_{\rho}: E^{+}\to E^{+}$ is liner, bounded and invertible (see ^[24]). Let $\widetilde{P}_{m}: E_m\to E_m^+$ be the orthogonal projection. Note that $L_{\rho}(E_{m})\subset E_{m}$ by Remark 2.1, then $\left(\widetilde{P}_{m}L_{\rho}\right)|_{E_{m}}: E_m^+\to E_m^+$ is also linear, bounded and invertible.

Then the assertion follows from Lemma 2.8 in ^[3].

Theorem 3.1. Assume $H$ satisfies (H1)–(H5), then there exists $\tilde{\tau} > 0$ such that for $\tau\geq\tilde{\tau}$, the system (1.3) possesses a nontrivial $2$-periodic brake orbit $z$ satisfying

Proof. The proof is standard, we proceed as that in ^[10,14].

For any $m\in\mathbb{N}$, Lemmas 3.1–3.4 show that $g_{m} = g|_{E_{m}}$ satisfies the hypotheses of the homological link Theorem 4.1.7 in ^[22], so $g_{m}$ possesses a critical point $z_{m}$ satisfying

By Lemma 3.1, when $\tau\geq\tau_0$, we may suppose $z_{m}\to z\in E$ as $m\to \infty$, then $g(z) > 0$ and $\nabla g(z) = 0$. By (H5), we see the critical point $z$ of $g$ is a classical nontrivial $2$-periodic brake orbit of the system (1.3).

Now we show (3.13) holds. Let $\mathcal{B}$ be the operator for $B(t) = \frac{\tau}{2}H^{\prime\prime}_{zz}(z(t))$ defined by (2.1), then

By (3.15), there exists $r_{0} > 0$ such that

where $d = \frac{1}{4}\| (B-A)^{\sharp} \|^{-1}$.

Hence, for $m$ large enough, there holds

For $x\in B_{r_{0}}\cap E_{m}$ and $w \in M_{d}^{+}(P_{m}(\mathcal{B}-\mathcal{A})P_{m}) \setminus \{0\}$, (3.16) implies that

Then

Note that

By (3.17), (3.18) and the link theorem 4.1.7 in ^[22], for large $m$, we have

Hence, we obtain $i_{L_{0}}(z, 1)\leq 0$.

Theorem 3.2. Assume $H$ satisfies (H1)–(H6), then there exists $\tilde{\tau}$ such that for $\tau\geq\tilde{\tau}$, the system (1.3) possesses a nontrivial brake orbit $z$ with minimal period $2$ or $1$.

Proof. The idea stems from ^[14], we proceed roughly.

For the nontrivial symmetric $2$-periodic brake orbit $z$ obtained in Theorem 3.1, assume its minimal period $\frac{2}{k}$ for some nonnegative integer $k$. Denote by $\gamma_{z, \frac{1}{k}}$ and $\gamma_{z}$ the corresponding symplectic path on the interval $[0, \frac{1}{k}]$ and $[0, 1]$ respectively, then $\gamma_{z} = \gamma^{k}_{z, \frac{1}{k}}$.

As shown in ^[14], we have the $L_{1}$-index estimate

By (H6), we see $B(t) = H''(z(t))$ is semipositive, Lemmas 2.1 and 2.2 and Eq (3.19) imply that

By Lemmas 2.2 and 2.3, we see

If $k$ is odd, by Lemma 2.4, we see

From (3.13), (3.20)–(3.22), we see $k = 1$.

If $k$ is even, If $k$ is even, by Lemma 2.4, we see

From (3.13), (3.20), (3.21) and (3.23), we have $k = 2$.

Acknowledgments

The first author is supported by the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (Grant No. 2021L377) and the Doctoral Scientific Research Foundation of Shanxi Datong University (Grant No. 2018-B-15). The authors sincerely thank the referees for their careful reading and valuable comments and suggestions.

Conflict of interest

The authors declare there is no conflicts of interest.