Stability switch and Hopf bifurcations for a diffusive plankton system with nonlocal competition and toxic effect

1.   Introduction

Fractional differential equations (FDEs) provide many mathematical models in physics, biology, economics, and chemistry, etc ^[1,2,3,4]. In fact, it consists of many integrals and derivative operators of non-integer orders, which generalize the theory of ordinary differentiation and integration. Hence, a more general approach is allowed to calculus and one can say that the aim of the FDEs is to consider various phenomena by studying derivatives and integrals of arbitrary orders. For intercalary specifics about the theory of FDEs, the readers are referred to the books of Kilbas et al.^[2] and Podlubny ^[4]. In the literature, several concepts of fractional derivatives have been represented, consisting of Riemann-Liouville, Liouville-Caputo, generalized Caputo, Hadamard, Katugampola, and Hilfer derivatives. The Hilfer fractional derivative ^[5] extends both Riemann-Liouville and Caputo fractional derivatives. For applications of Hilfer fractional derivatives in mathematics and physics, etc see ^{[6,7,8,9,10,11]}. For recent results on boundary value problems for fractional differential equations and inclusions with the Hilfer fractional derivative see the survey paper by Ntouyas ^[12]. The $\psi$-Riemann-Liouville fractional integral and derivative operators are discussed in ^[1], while the $\psi$-Hilfer fractional derivative is discussed in ^[13]. Recently, the notion of a generalized proportional fractional derivative was introduced by Jarad et al. ^[14,15,16]. For some recent results on fractional differential equations with generalized proportional derivatives, see ^[17,18].

In ^[19], an existence result was proved via Krasnosel'ski$\breve{{\rm{i}}}$'s fixed-point theorem for the following sequential boundary value problem of the form

where ${}^{H}\mathbb{D}^{\omega, \varsigma, \psi}$ indicates the $\psi$-Hilfer fractional derivative of order $\omega\in\{\alpha, \beta\}$, with $0 < \alpha\leq 1$, $1 < \beta\leq 2$, $0\leq\varsigma < 1$, $\mathbb{I}^{\nu_{i}; \psi}$ is the $\psi$-Riemann–Liouville fractional integral of order $\nu_{i} > 0$, for $i = 1, 2, \dots, n$, $h_{i}\in C([0, 1]\times \mathbb{R}, \mathbb{R})$, for $i = 1, 2, \dots, n$, $\phi\in C([0, 1]\times \mathbb{R}, \mathbb{R}\setminus\{0\})$, $\Upsilon\in C([0, 1]\times \mathbb{R}, \mathbb{R})$, $\tau\in \mathbb{R}$ and $\zeta\in (a, b)$. In ^[16], the consideration of Hilfer-type generalized proportional fractional derivative operators was initiated.

Coupled systems of fractional order are also significant, as such systems appear in the mathematical models in science and engineering, such as bio-engineering ^[20], fractional dynamics ^[21], financial economics ^[22], etc. Coupled systems of FDEs with diverse boundary conditions have been the focus of many researches. In ^[23], the authors studied existence and Ulam-Hyers stability results of a coupled system of $\psi$-Hilfer sequential fractional differential equations. Existence and uniqueness results are derived in ^[24] for a coupled system of Hilfer-Hadamard fractional differential equations with fractional integral boundary conditions. Recently, in ^[25] a coupled system of nonlinear fractional differential equations involving the $(k, \psi)$-Hilfer fractional derivative operators complemented with multi-point nonlocal boundary conditions were discussed. Moreover, Samadi et al. ^[26] have considered a coupled system of Hilfer-type generalized proportional fractional differential equations.

In this article, motivated by the above works, we study a coupled system of $\psi$-Hilfer sequential generalized proportional FDEs with boundary conditions generated by the problem (1.1). More precisely, we consider the following coupled system of nonlinear proportional $\psi$-Hilfer sequential fractional differential equations with multi-point nonlocal boundary conditions of the form

where ${}^{H}D^{\nu, \vartheta_{1}; \varsigma; \psi}$ denotes the $\psi$-Hilfer generalized proportional derivatives of order $\nu\in \{\nu_1, \nu_2, \nu_3, \nu_4\},$ with parameters $\vartheta_{l}$, $0\leq \vartheta_{l}\leq 1$, $l\in\{1, 2, 3, 4\}$, $\psi$ is a continuous function on $[t_{1}, t_{2}],$ with ${\psi}'(w) > 0,$ $^{p}I^{\eta, \varsigma, \psi}$ is the generalized proportional integral of order $\eta > 0$, $\eta\in \{\eta_{i}, \eta_{j}\}$, $\theta_{1}, \theta_{2} \in \mathbb{R}$, $\xi_{1}, \xi_{2}\in [t_{1}, t_{2}]$, $\Phi_{1}, \Phi_{2}\in C([t_{1}, t_{2}]\times \mathbb{R}\times \mathbb{R}, \mathbb{R}\setminus\{0\})$ and $H_{i}, G_{j}, \Upsilon_{1}, \Upsilon_{2}\in C([t_{1}, t_{2}]\times \mathbb{R}\times \mathbb{R}, \mathbb{R})$, for $i = 1, 2, \dots, n$ and $j = 1, 2, \dots, m$.

We emphasize that:

● We study a general system involving $\psi$-Hilfer proportional fractional derivatives.

● Our equations contain fractional derivatives of different orders as well as sums of fractional integrals of different orders.

● Our system contains nonlocal coupled boundary conditions.

● Our system covers many special cases by fixing the parameters involved in the problem. For example, taking $\psi(w) = w$, it will reduce to a coupled system of Hilfer sequential generalized proportional FDEs with boundary conditions, while if $\varsigma = 1$, it reduces to a coupled system of $\psi$-Hilfer sequential FDEs. Besides, by taking $\Phi_{1}, \Phi_{2} = 1$ in the problem (1.2), then we obtain the following new coupled system of the form:

In obtaining the existence result of the problem (1.2), first the problem (1.2) is converted into a fixed-point problem and then a generalization of Krasnosel'ski$\breve{{\rm{i}}}$'s fixed-point theorem due to Burton is applied.

The structure of this article has been organized as follows: In Section $2$, some necessary concepts and basic results concerning our problem are presented. The main result for the problem (1.2) is proved in Section $3$, while Section $4$ contains an example illustrating the obtained result.

2.   Preliminaries

In this section, we summarize some known definitions and lemmas needed in our results.

Definition 2.1. ^[17,18] Let $\varsigma\in (0, 1]$ and $\nu > 0$. The fractional proportional integral of order $\nu$ of the continuous function $\mathfrak{F}$ is defined by

Definition 2.2. ^[17,18] Let $\varsigma\in (0, 1]$, $\nu > 0$, and $\psi(w)$ is a continuous function on $[t_{1}, t_{2}],$ ${\psi}'(w) > 0$. The generalized proportional fractional derivative of order $\nu$ of the continuous function $\mathfrak{F}$ is defined by

where $n = [\rho]+1$ and $[\nu]$ denotes the integer part of the real number $\nu,$ where $D^{n, \varsigma, \psi} = \underbrace{D^{\varsigma, \psi}\cdots D^{\varsigma, \psi}}_{n-times}$.

Now the generalized Hilfer proportional fractional derivative of order $\nu$ of function $\mathfrak{F}$ with respect to another function $\psi$ is introduced.

Definition 2.3. ^[27] Let $\varsigma\in (0, 1]$, $\mathfrak{F}, \psi\in C^{m}([t_{1}, t_{2}], \mathbb{R})$ in which $\psi$ is positive and strictly increasing with ${\psi}^{'}(w)\neq 0$ for all $w\in [t_{1}, t_{2}]$. The $\psi$-Hilfer generalized proportional fractional derivative of order $\nu$ and type $\vartheta$ for $\mathfrak{F}$ with respect to another function $\psi$ is defined by

where $n-1 < \nu < n$ and $0\leq\vartheta\leq1$.

Lemma 2.4. ^[27] Let $m-1 < \nu < m, n\in N$, $0 < \varsigma\leq 1$, $0\leq \vartheta \leq 1$ and $m-1 < \gamma < m$ such that $\gamma = \nu + m\vartheta-\nu\vartheta$. If $\mathfrak{F}\in C([t_{1}, t_{2}], \mathbb{R})$ and $^{p}I^{(m -\gamma, \varsigma, \psi)}\mathfrak{F}\in C^{m}([t_{1}, t_{2}], \mathbb{R})$, then

To prove the main result we need the following lemma, which concerns a linear variant of the $\psi$-Hilfer sequential proportional coupled system (1.2). This lemma plays a pivotal role in converting the nonlinear problem in system (1.2) into a fixed-point problem.

Lemma 2.5. Let $0 < \nu_{1}, \nu_{3}\leq 1$, $1 < \nu_{2}, \nu_{4}\leq 2$, $0\leq \vartheta_{i} \leq1$, $\gamma_{i} = \nu_{i} + \vartheta_{i}(1-\nu_{i})$, $i = 1, 3$ and $\gamma_{j} = \nu_{j} + \vartheta_{j}(2-\nu_{j})$, $j = 2, 4$, $\Theta = M_{1}N_{2}-M_{2}N_{1}\neq 0,$ $\psi$ is a continuous function on $[t_{1}, t_{2}],$ with ${\psi}'(w) > 0,$ and $Q_{1}, Q_{2}\in C([t_{1}, t_{2}], \mathbb{R}),$ $\Phi_{1}, \Phi_{2}\in C([t_{1}, t_{2}]\times \mathbb{R}\times \mathbb{R}, \mathbb{R}\setminus\{0\})$ and $H_{i}, G_{j}, Q_{1}, Q_{2}\in C([t_{1}, t_{2}]\times \mathbb{R}\times \mathbb{R}, \mathbb{R})$, for $i = 1, 2, \dots, n$ and $j = 1, 2, \dots, m,$ and $^{p}I^{(1 -\gamma_i, \varsigma, \psi)}Q_j\in C^{m}([t_{1}, t_{2}], \mathbb{R}), i = 1, 2, 3, 4, j = 1, 2$. Then the pair $(p_{1}, p_{2})$ is a solution of the system

if and only if

and

where

Proof. Due to Lemma 2.4 with $m = 1$, we get

where $c_{0}, d_{0}\in \mathbb{R}.$ Now applying the boundary conditions

we get $c_{0} = d_{0} = 0$. Hence

Now, by taking the operators ${}^{p}I^{\nu_{2}, \varsigma, \psi}$ and ${}^{p}I^{\nu_{4}, \varsigma, \psi}$ into both sides of (2.5) and using Lemma 2.4, we get

Applying the conditions $p_{1}(t_{1}) = p_{2}(t_{1}) = 0$ in (2.6), we get $c_{2} = d_{2} = 0$ since $\gamma_{2}\in [\nu_{2}, 2]$ and $\gamma_{4}\in [\nu_{4}, 2].$ Thus we have

In view of (2.7) and the conditions $p_{1}(t_{2}) = \theta_{1}p_{2}(\xi_{1})$ and $p_{2}(t_{2}) = \theta_{2}p_{1}(\xi_{2})$, we get

and

Due to (2.3), (2.8), and (2.9), we have

where

By solving the above system, we conclude that

Replacing the values $c_{1}$ and $d_{1}$ in Eq (2.7), we obtain the solutions (2.1) and (2.2). The converse is obtained by direct computation. The proof is complete. □

3.   An existence result

Let $\mathbb{Y} = C([t_{1}, t_{2}], \mathbb{R}) = \{p:[t_{1}, t_{2}]\longrightarrow \mathbb{\mathbb{R}}\; \mbox{is continuous}\}.$ The space $\mathbb{Y}$ is a Banach space with the norm $\| p \| = \sup_{w\in [t_{1}, t_{2}]}| p(w) |$. Obviously, the space $(\mathbb{Y}\times \mathbb{Y}, \| (p_{1}, p_{2}) \|)$ is also a Banach space with the norm $\|(p_{1}, p_{2})\| = \| p_{1} \| + \| p_{2} \|.$

Due to Lemma 2.5, we define an operator $\mathbb{V}: \mathbb{Y}\times \mathbb{Y}\to \mathbb{Y}\times \mathbb{Y}$ by

where

and

To prove our main result we will use the following Burton's version of Krasnosel'ski$\breve{{\rm{i}}}$'s fixed-point theorem.

Lemma 3.1. ^[28] Let $S$ be a nonempty, convex, closed, and bounded set of a Banach space $(X, \|\cdot\|)$ and let $\mathcal{A} : \mathbb{X} \to \mathbb{X}$ and $\mathcal{B} : S \to \mathbb{X}$ be two operators which satisfy the following:

(i) $\mathcal{A}$ is a contraction,

(ii) $\mathcal{B}$ is completely continuous, and

(iii) $x = \mathcal{A}x+\mathcal{B}y, \forall y\in S\Rightarrow x\in S.$

Then there exists a solution of the operator equation $x = \mathcal{A}x+\mathcal{B}x.$

Theorem 3.2. Assume that:

$(H_{1})$ The functions $\Phi_k : [t_{1}, t_{2}]\times\mathbb{R}^2 \to \mathbb{R}\setminus\{0\},$ $\Upsilon_k: [t_{1}, t_{2}]\times\mathbb{R}^2 \to \mathbb{R}$ for $k = 1, 2$ and $h_{i}, g_j : [t_{1}, t_{2}]\times\mathbb{R}^2 \to \mathbb{R}$ for $i = 1, 2, \ldots, n, \; j = 1, 2, \ldots, m$, are continuous and there exist positive continuous functions $\phi_k$, $\omega_k:[t_{1}, t_{2}]\to \mathbb{R},$ $k = 1, 2,$ $h_{i}: [t_{1}, t_{2}]\to \mathbb{R}$, $g_j:[t_{1}, t_{2}]\to \mathbb{R}$ $i = 1, 2, \ldots, n\; j = 1, 2, \ldots, m$, with bounds $\|\phi_k\|,$ $\|\omega_k\|,$ $k = 1, 2,$ and $\|h_{i}\|$, $i = 1, 2, \ldots, m$, $\|g_j\|, \; j = 1, 2, \ldots, m,$ respectively, such that

for all $w\in [t_{1}, t_{2}]$ and $u_{i}, \overline{u}_{i}\in \mathbb{R}$, $i = 1, 2.$

$(H_{2})$ There exist continuous functions $F_{k}, L_{k}, k = 1, 2,$ $\lambda_i, \mu_{j}, i = 1, 2, \ldots, n, \; j = 1, 2, \ldots, m$ such that

for all $w\in [t_{1}, t_{2}]$ and $u_{1}, u_{2} \in \mathbb{R}$.

$(H_3)$ Assume that

where $\|F_k\| = \sup_{t\in [t_1, t_2]}|F_k(t)|,$ $\|L_k\| = \sup_{t\in [t_1, t_2]}, k = 1, 2,$ $\|\lambda_i\| = \sup_{t\in [t_1, t_2]},$ $i = 1, 2, \ldots, n,$ and $\|\mu_j\| = \sup_{t\in [t_1, t_2]},$ $j = 1, 2, \ldots, m.$

Then the $\psi$-Hilfer sequential proportional coupled system (1.2) has at least one solution on $[t_1, t_2].$

Proof. First, we consider a subset $S$ of $\mathbb{Y}\times \mathbb{Y}$ defined by $S = \{(p_1, p_2)\in \mathbb{Y}\times \mathbb{Y}: \|(p_1, p_2)\|\le r\}$, where $r$ is given by

where

and

Let us define the operators:

and

Then we have

and

Also, we obtain

and

Moreover, we have

and

Finally, we get

and

Now we split the operator $\mathbb{V}$ as

with

and

In the following, we will show that the operators $\mathbb{V}_{1}$ and $\mathbb{V}_{2}$ fulfill the assumptions of Lemma 3.1. We divide the proof into three steps:

Step 1. The operators $\mathbb{V}_{1, 1}$ and $\mathbb{V}_{2, 1}$ are contraction mappings. For all $(p_{1}, p_{2}), (\overline{p}_{1}, \overline{p}_{2})\in \mathbb{Y}\times \mathbb{Y}$ we have

Similarly we can find

Consequently, we get

which means that $(\mathbb{V}_{1, 1}, \mathbb{V}_{2, 1})$ is a contraction.

Step 2. The operator $\mathbb{V}_{2} = (\mathbb{V}_{1, 2}, \mathbb{V}_{2, 2})$ is completely continuous on $S.$ For continuity of $\mathbb{V}_{1, 2}$, take any sequence of points $(p_n, q_n)$ in $S$ converging to a point $(p, q) \in S.$ Then, by the Lebesgue dominated convergence theorem, we have

for all $w\in [t_1, t_2].$ Similarly, we prove $\lim_{n\to \infty}\mathbb{V}_{2, 2}(p_n, q_n)(w) = \mathbb{V}_{2, 2}(p, q)(w)$ for all $w\in [t_1, t_2].$ Thus $\mathbb{V}_{2}(p_n, q_n) = (\mathbb{V}_{1, 2}(p_n, q_n), \mathbb{V}_{2, 2}(p_n, q_n))$ converges to $\mathbb{V}_{2}(p, q)$ on $[t_1, t_2],$ which shows that $\mathbb{V}_{2}$ is continuous.

Next, we show that the operator $(\mathbb{V}_{1, 2}, \mathbb{V}_{2, 2})$ is uniformly bounded on $S.$ For any $(p_1, p_2)\in S$ we have

Similarly we can prove that

Therefore $\|\mathbb{V}_{1, 2}\|+\|\mathbb{V}_{2, 2}\|\le \Lambda_1+\Lambda_2, (p_1, p_2)\in S,$ which shows that the operator $(\mathbb{V}_{1, 2}, \mathbb{V}_{2, 2})$ is uniformly bounded on $S.$ Finally we show that the operator $(\mathbb{V}_{1, 2}, \mathbb{V}_{2, 2})$ is equicontinuous. Let $\tau_1 < \tau_2$ and $(p_1, p_2)\in S.$ Then, we have

where

As $\tau_2-\tau_1\to 0$, the right-hand side of the above inequality tends to zero, independently of $(p_1, p_2)$. Similarly we have $|\mathbb{V}_{2, 2}(p_1, p_2)(\tau_2)-\mathbb{V}_{2, 2}(p_1, p_2)(\tau_1)|\to 0$ as $\tau_2-\tau_1\to 0.$ Thus $(\mathbb{V}_{1, 2}, \mathbb{V}_{2, 2})$ is equicontinuous. Therefore, it follows by the Arzelá-Ascoli theorem that $(\mathbb{V}_{1, 2}, \mathbb{V}_{2, 2})$ is a completely continuous operator on $S.$

Step 3. We show that the third condition (iii) of Lemma 3.1 is fulfilled. Let $(p_1, p_2)\in \mathbb{Y}\times \mathbb{Y}$ be such that, for all $(\overline{p}_1, \overline{p}_2)\in S$

Then, we have

In a similar way, we find

Adding the previous inequalities, we obtain

As $\|(p_1, p_2)\| = \|p_1\|+\|p_2\|,$ we have that $\|(p_1, p_2)\|\le r$ and so condition (iii) of Lemma 3.1 holds.

By Lemma 3.1, the $\psi$-Hilfer sequential proportional coupled system (1.2) has at least one solution on $[t_1, t_2].$ The proof is finished. □

4.   An example

Let us consider the following coupled system of nonlinear sequential proportional Hilfer fractional differential equations with multi-point boundary conditions:

where

Next, we can choose $\nu_{1} = 1/3$, $\nu_{2} = 5/4$, $\nu_{3} = 2/3$, $\nu_{4} = 7/4$, $\vartheta_{1} = 1/5$, $\vartheta_{2} = 2/5$, $\vartheta_{3} = 3/5$, $\vartheta_{4} = 4/5$, $\varsigma = 3/7$, $\psi(w): = \log w = \log_e w$, $t_1 = 1/2$, $t_2 = 7/2$, $\theta_{1} = 2/5$, and $\theta_{2} = 2/3$. Then, we have $\gamma_1 = 7/15$, $\gamma_{2} = 31/20$, $\gamma_{3} = 13/15$, $\gamma_{4} = 39/20$, $M_1\approx0.1930945138$, $M_2\approx0.2307306625$, $N_1\approx0.1816223751$, $N_2\approx0.3208292984$, and $\Theta\approx0.02004452646$. Now, we analyse the nonlinear functions in the fractional integral terms. We have

and

from which $h_i(w) = 1/(i(w+i^2))$ and $g_j(w) = 1/(j(w^2+j^2))$, respectively. Both of them are bounded as

Therefore $\lambda_i(w) = 2/(w+i^2)$ and $\mu_{j} = 2/(w^2+j^2)$. Moreover, we have

and

For the two non-zero functions $\Phi_1$ and $\Phi_2$ we have

from which we get $\|\phi_1\| = 1/26000$, $\|\phi_{2}\| = 1/25000$, $\|F_1\| = 1/10400$, $\|F_2\| = 9/100000,$ by setting $\phi_{1}(w) = 1/(100(10w+255))$, $\phi_{2}(w) = 2/(5(2w+99)^2)$, $F_1(w) = 1/(40(10w+255)),$ and $F_2(w) = 9/(10(2w+99)^2)$, respectively.

Finally, for the nonlinear functions of the right sides in problem (4.1) we have

which give $\omega_1(w) = 1/(2(\sqrt{w}+1))$, $\omega_2(w) = 1/(2(w^2+4))$ and

and

Therefore, using all of the information to compute a constant $K$ in assumption $(H_3)$ of Theorem 3.2, we obtain

Hence, the given coupled system of nonlinear proportional Hilfer-type fractional differential equations with multi-point boundary conditions (4.1), satisfies all assumptions in Theorem 3.2. Then, by its conclusion, there exists at least one solution $(p_1, p_2)(w)$ to the problem (4.1) where $w\in[1/2, 7/2]$.

5.   Conclusions

In this paper, we have presented the existence result for a new class of coupled systems of $\psi$-Hilfer proportional sequential fractional differential equations with multi-point boundary conditions. The proof of the existence result was based on a generalization of Krasnosel'ski$\breve{{\rm{i}}}$'s fixed-point theorem due to Burton. An example was presented to illustrate our main result. Some special cases were also discussed. In future work, we can implement these techniques on different boundary value problems equipped with complicated integral multi-point boundary conditions.

Use of AI tools declaration

The authors declare that they have not used artificial intelligence (AI) tools in the creation of this article.

Acknowledgments

This research was funded by the National Science, Research and Innovation Fund (NSRF) and King Mongkut's University of Technology North Bangkok with contract no. KMUTNB-FF-66-11.

Conflict of interest

Professor Sotiris K. Ntouyas is an editorial board member for AIMS Mathematics and was not involved in the editorial review or the decision to publish this article. The authors declare no conflicts of interest.