
In this paper, we studied a generalized reaction-diffusion system that models predator-prey dynamics, incorporating prey-taxis along with a hunting cooperation effect and a logistic source for the predator population, subject to homogeneous Neumann boundary conditions. This system describes a predator-prey interaction in which the prey, via a prey-taxis mechanism, employ group defense strategies against their predators, while the predators, through their functional response and net growth rate, not only cooperate in hunting these defended prey but also exhibit logistic growth dynamics, thereby ensuring the self-regulation of the predator population. We established that solutions to the time- and space-dependent system exhibiting such ecological characteristics exist globally and remain bounded within a one-dimensional spatial domain. Furthermore, using global bifurcation theory, we proved the existence of nonconstant positive steady states. A key ingredient in this analysis is the derivation of uniform a priori estimates for positive steady-state solutions, which play a crucial role in controlling the global solution branches.
Citation: Kimun Ryu, Wonlyul Ko. Global existence of classical solutions and steady-state bifurcation in a prey-taxis predator-prey system with hunting cooperation and a logistic source for predators[J]. Electronic Research Archive, 2025, 33(6): 3811-3833. doi: 10.3934/era.2025169
[1] | Xuemin Fan, Wenjie Zhang, Lu Xu . Global dynamics of a predator-prey model with prey-taxis and hunting cooperation. Electronic Research Archive, 2025, 33(3): 1610-1632. doi: 10.3934/era.2025076 |
[2] | Ailing Xiang, Liangchen Wang . Boundedness of a predator-prey model with density-dependent motilities and stage structure for the predator. Electronic Research Archive, 2022, 30(5): 1954-1972. doi: 10.3934/era.2022099 |
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[8] | Yichao Shao, Hengguo Yu, Chenglei Jin, Jingzhe Fang, Min Zhao . Dynamics analysis of a predator-prey model with Allee effect and harvesting effort. Electronic Research Archive, 2024, 32(10): 5682-5716. doi: 10.3934/era.2024263 |
[9] | San-Xing Wu, Xin-You Meng . Hopf bifurcation analysis of a multiple delays stage-structure predator-prey model with refuge and cooperation. Electronic Research Archive, 2025, 33(2): 995-1036. doi: 10.3934/era.2025045 |
[10] | Yuan Tian, Yang Liu, Kaibiao Sun . Complex dynamics of a predator-prey fishery model: The impact of the Allee effect and bilateral intervention. Electronic Research Archive, 2024, 32(11): 6379-6404. doi: 10.3934/era.2024297 |
In this paper, we studied a generalized reaction-diffusion system that models predator-prey dynamics, incorporating prey-taxis along with a hunting cooperation effect and a logistic source for the predator population, subject to homogeneous Neumann boundary conditions. This system describes a predator-prey interaction in which the prey, via a prey-taxis mechanism, employ group defense strategies against their predators, while the predators, through their functional response and net growth rate, not only cooperate in hunting these defended prey but also exhibit logistic growth dynamics, thereby ensuring the self-regulation of the predator population. We established that solutions to the time- and space-dependent system exhibiting such ecological characteristics exist globally and remain bounded within a one-dimensional spatial domain. Furthermore, using global bifurcation theory, we proved the existence of nonconstant positive steady states. A key ingredient in this analysis is the derivation of uniform a priori estimates for positive steady-state solutions, which play a crucial role in controlling the global solution branches.
In [1], we proposed and studied the following predator-prey reaction-diffusion system, which incorporates a hunting cooperation functional response and prey-taxis:
{ut−τΔu=g(u)−vf(u,v)in (0,∞)×Ω,vt−Δv−χ∇⋅(q(v)∇u)=βvf(u,v)−γ(v)in (0,∞)×Ω,∂nu=∂nv=0on (0,∞)×∂Ω,u(0,⋅)=u0≥0, v(0,⋅)=v0≥0in Ω. | (1.1) |
Here, u(t,x) and v(t,x) represent the population densities of prey and predators, respectively, at time t and spatial location x∈Ω. The spatial domain Ω⊆RN, where N≥1, is bounded and has a smooth boundary, denoted by ∂Ω. Furthermore, n(x) denotes the outward unit normal vector on ∂Ω, and ∂n=∂/∂n. In this system, τ>0 is the constant diffusion rate of the prey species, and the term −χ∇⋅(q(v)∇u) stands for (repulsive) prey-taxis, where q(v) is the sensitivity function and χ>0 is the chemotaxis coefficient. Furthermore, the function g(u) characterizes the intrinsic growth rate of the prey population, while f(u,v) represents the functional response, indicating the rate at which predators hunt and consume their prey. The constant β>0 denotes the conversion rate associated with predation, and γ(v) indicates the net growth rate of predators in the absence of prey. Finally, the initial data u0 and v0 are assumed to satisfy
u0,v0∈W1,p(Ω) for some p>N, u0≢0, v0≢0 in Ω. |
Ecologically, the constant diffusion rate τ quantifies the random dispersal of the prey species throughout the domain Ω. However, due to the prey-taxis term, predators tend to move in the direction opposite to the increasing prey density gradient, resulting in a scenario in which prey exhibit a group defense strategy against predators (e.g., see [2,3]). An example of such a predator-prey interaction involving group defense through the prey-taxis mechanism, as described in (1.1), can be found in [4]. Furthermore, it is biologically realistic for predators to exhibit a limited response when prey densities are low (see [5,6]). Accordingly, the sensitivity function of predators, represented as q(v), is expected to be bounded. The intrinsic growth rate of the prey population, denoted as g(u), is typically influenced by resource limitations and is often expressed as follows:
ru(1−uK) or r(1−uK), |
where r and K are positive constants. Regarding the capturing rate, the functional response f(u,v) generally increases with prey density. In ecosystems with cooperative hunting, higher predator densities enhance the capture rate; however, interference among predators (see [7]) limits this effect. Consequently, f(u,v) increases with both predator and prey densities, but remains bounded. Representative forms of hunting-cooperation functional responses (e.g., see [1,7,8,9,10,11,12,13]) include
Ce0uv1+h0Ce0uv, Ce0u(1+av)1+h0Ce0u(1+av), and Ce0u2(1+av)1+h0Ce0u2(1+av), |
where a, e0, h0, and C are positive constants (for their biological meanings, see [7,8,12,13,14]). The net growth rate of predators, denoted as γ(v), may be influenced by the predator death rate or by intraspecific competition. Accordingly, prototype forms for γ are provided by
vMγ and Mγv+γ∗v2, |
where Mγ and γ∗ are positive constants representing the intrinsic death rate and the intraspecific competition rate, respectively. To incorporate these ecological backgrounds in the functions g, f, γ, and q [1], assumed that there exist positive constants K, Mg, Mf, Mγ, and Mq such that the following general hypotheses are satisfied:
(H1) g∈C1([0,∞),R), g(0)≥0, g(K)=0, 0<g(u)≤Mg for all 0<u<K, while g(u)<0 for any u>K.
(H2) f∈C1([0,∞)2,[0,∞)), f(u,0)≥0, f(0,v)=0, fu(u,v)>0, and fv(u,v)>0 for any u, v>0, and f(u,v)≤Mf for all u, v≥0.
(H3) γ∈C1([0,∞),[0,∞)), γ(0)=0, γ′(0)=Mγ, γ(v)≥Mγv, and γ′(v)>0 for any v≥0.
(H4) q∈C1([0,∞),[0,∞)), q(0)=0, and q(v)≤Mqv for all v≥0.
Under these assumptions, in [1], we established the global existence of classical solutions to system (1.1) for all N≥1 when the chemotaxis coefficient χ is sufficiently small. To achieve this, we employed Amann's local existence theorem [15,16], along with various estimates for the diffusion semigroup and the Gagliardo-Nirenberg inequality (e.g., [17,18,19,20]). Furthermore, we derived a weighted integral estimate using a new weight function, which subsequently yielded an explicit upper bound for χ ensuring the existence of bounded solutions.
Not only for Keller–Segel chemotaxis models without reaction terms but also for predator-prey models with prey-taxis, the global existence and boundedness of solutions have been established under either a technical truncation assumption (see [21,22,23]) or a smallness assumption on the prey-taxis coefficient (see [1,24]). Such assumptions are often employed to prevent population overcrowding. The global existence of solutions to PDEs with chemotaxis terms has recently emerged as one of the most actively studied topics in biological mathematics. Our earlier results in [1] align with this trend. In particular, for studies on predator-prey models incorporating prey-taxis, we refer the reader to [1,18,24] and the references therein. This raises a fundamental question: Can global existence of solutions be ensured without imposing any restrictive assumptions? In addressing this, several notable results have demonstrated global existence for all χ>0, without requiring smallness or truncation conditions (e.g., see [4,18,25,26,27,28,29]).
In [25], a prey-taxis system with the ratio-dependent functional response (uu+av)v (for a contant a>0) was studied for all χ≥0 and N≥1. The key inequality (1u+av)v≤1a played a crucial role in obtaining the desired result. The models in [4,26,28,29] incorporate Lotka–Volterra-type interactions among species. In [18], the authors considered a prey-dependent functional response f(u) satisfying the Holling type-Ⅱ condition, combined with a logistic-type source in the predator equation. They established the global existence of classical solutions for all χ>0, but only for the case N=2. Meanwhile, [27] addressed a chemotaxis model in one spatial dimension (N=1). These studies collectively highlight that, in ensuring global existence of solutions for prey-taxis models with arbitrary χ>0, the structure of the functional response plays a crucial role. In particular, when attempting to apply an entropy-like equality (cf. [30]) as was done in [18] to our system (1.1) with a functional response f(u,v) satisfying (H2), one encounters difficulties. This is due to the fact that f depends on both u and v, and the quantity f(u,v)u is not necessarily bounded. These aspects impose certain limitations on the applicability of various theoretical approaches, including entropy-based methods. In this context, the analysis of prey-taxis models with functional responses satisfying (H2) presents a certain level of novelty. To overcome the associated difficulties and eliminate the need for a smallness condition on the prey-taxis sensitivity coefficient χ, we begin our study under structural assumptions that incorporate both a logistic-type source and a one-dimensional spatial setting. These assumptions allow for sufficient analytical control, facilitating the derivation of the required a priori estimates.
We note that the question of whether similar results can be obtained without such restrictions (e.g., in higher dimensions or without a logistic source) remains largely open. As seen in the works cited above, the global existence and boundedness of solutions to prey-taxis predator-prey systems is a topic of active ongoing research. Taken together, these studies suggest that either the spatial dimension N or the presence of a logistic source plays an essential role in the derivation of key estimates, particularly those involving the Gagliardo–Nirenberg inequality. Motivated by this, we study global existence and boundedness in a one-dimensional model with a suitably structured logistic source. In particular, we focus on the impact of the predator growth term and explore whether a refined assumption on γ(v) can enhance the analysis in the one-dimensional case. To this end, we replace assumption (H3) with the following, more stringent condition:
(H3*) γ∈C1([0,∞),[0,∞)), γ(0)=0, γ′(0)=Mγ, γ(v)≥Mγv+γ∗vα, and γ′(v)>0 for some α>1 and all v≥0.
We note that the term γ∗vα in (H3*) contributes a self-limiting mechanism to the predator equation. While not logistic in the classical algebraic sense (e.g., of the form v(1−v)), this higher-order damping term serves a similar ecological role by suppressing unbounded population growth at high densities. For this reason, we refer to it as a logistic-type source in the biological modeling. Additionally, prey-taxis predator-prey models featuring similar generalized logistic sources can be found in [29] (see also [28]).
Having established the motivation for incorporating (H3*) into our analysis, we now turn to the main objective of this paper. As a continuation of the discussion presented in Remark 2.2 of [1], we aim to prove the global existence and boundedness of solutions to system (1.1) for any χ>0 in the one-dimensional case (i.e., N=1). Accordingly, we present our main result for system (1.1) as follows.
Theorem 1.1. Assume that assumptions (H1), (H2), (H3*), and (H4) hold and that N=1. Let u0, v0∈W1,p(Ω) for some p>1, where u0, v0≥0(≢0) in Ω. Then, system (1.1) admits a unique, nonnegative, and global classical solution (u,v), which satisfies the properties
u, v∈C([0,∞);W1,p(Ω))∩C2,1((0,∞)ׯΩ), |
and u>0 in (0,∞)ׯΩ. Moreover, the solution is bounded in the sense that there exists a constant C>0, independent of t, such that
‖u(t,⋅)‖W1,∞(Ω)+‖v(t,⋅)‖L∞(Ω)≤C for all t>0. |
Having established the global existence and boundedness of classical solutions in one dimension through Theorem 1.1, we now turn our attention to the steady states of (1.1). In particular, an important question arises in the context of reaction-diffusion predator-prey systems with prey-taxis: under what conditions can spatial patterns emerge in the steady-state regime? This phenomenon, commonly referred to as pattern formation, is of great biological and mathematical interest (see, for instance, [4,9,26,31,32,33,34]). In the context of system (1.1), the presence of prey-taxis and nonlinear interactions through cooperative hunting and logistic-type regulation can induce instabilities in constant steady states, potentially leading to the emergence of spatially nonhomogeneous solutions. Such phenomena are often studied via bifurcation theory (e.g., [26,32,33,34]), particularly when investigating how variations in parameters such as the prey-taxis sensitivity χ influence the structure of steady states. Motivated by this, we analyze the associated steady-state elliptic system, which governs the equilibrium configurations of parabolic system (1.1). By employing global bifurcation theory, we rigorously demonstrate the existence of nonconstant positive steady states bifurcating from spatially homogeneous equilibria. A key element of this analysis is the derivation of a priori estimates for the steady-state solutions, which are essential for controlling the global bifurcation branches and ensuring the applicability of the theory.
As the second main result of this paper, we present a result on the global bifurcation of positive solutions to the following elliptic system, which corresponds to the steady states of (1.1):
{−τΔu=g(u)−vf(u,v)in Ω,−Δv−χ∇⋅(q(v)∇u)=βvf(u,v)−γ(v)in Ω,∂nu=∂nv=0on ∂Ω. | (1.2) |
The bifurcation branch emanates from a positive constant steady state of the system. Accordingly, we first ensure the existence of such a constant solution by referring to [1, Theorem 2.3].
Proposition 1.2. Assume that assumptions (H1), (H2), (H3*), and (H4) hold. Assume, additionally, that
(H1b) g′(K)<0 and g′(0)>0 hold when g(0)=0; g′(K)<0 holds when g(0)>0.
Let
ξ(u)=γ−1(βg(u)) and H(u)=g(u)−f(u,ξ(u))ξ(u). |
Then (1.1) has at least one steady state of constant coexistence if either one of the following is fulfilled:
(i)H′(K)>0 (i.e., βf(K,0)−γ′(0)>0);(ii)H′(K)≤0 and H(M∗)<0 for some M∗∈(0,K). | (1.3) |
For convenience in stating the following theorem, we now prepare the following notations: Let
0=μ0<μ1≤μ2<⋯<μi<⋯ and lim |
be all eigenvalues of the problem in and on . Additionally, let denote the normalized eigenfunction corresponding to for . In particular, when , we know that and . Furthermore, when (1.2) possesses a positive constant solution , we let
where
Theorem 1.3. Assume that assumptions (H1), (H2), (H3*), (H4), and (H1b) hold and that . Furthermore, suppose that , and that one of the conditions in (1.3) is satisfied. Assume additionally that
(H5) for all positive constant solutions of (1.2).
Let
such that
Then the following results hold:
(i) There exists a positive constant such that nonconstant positive solutions of (1.2) lie on a smooth curve
near at , where is a continuous function such that , , and for some smooth functions and satisfying .
(ii) The curve is contained in a global branch of positive solutions of (1.2). Moreover, either joins with , or joins to in .
The following is a brief overview of the paper's structure. In Section 2, we present several preliminary results that are essential for the proof of Theorem 1.1. These include a local existence theorem for system (1.1) and various estimates related to its solutions. In particular, to establish -estimates for , we employ the Moser-type iteration method [35] as one of our primary analytical tools. Subsequently, we apply all the previously derived results to complete the proof of Theorem 1.1. Next, using global bifurcation theory [36], we prove the second main result, Theorem 1.3, which establishes that a nonconstant positive solution branch of (1.2) bifurcates from a positive constant equilibrium. As a preliminary step in this analysis, we derive a priori estimates for positive solutions to (1.2).
The first part of this section is devoted to the proof of Theorem 1.1. To begin, we present a result obtained from [1] stating that (1.1) admits a local classical solution. This can be proven using Amann's local existence results [15,16] (e.g., see [18,24,37] for similar applications).
Lemma 2.1. Assume that (H1), (H2), (H3*), and (H4) hold, and suppose , with some , where , and , in . Then, there exists a maximal-existence-time such that system (1.1) has a unique classical solution , where , are nonnegative functions satisfying . Moreover, the criterion for extensibility holds:
When is a solution of system (1.1), [1] established an -norm bound of by applying both the comparison principle and the strong maximum principle, as well as an -norm bound of through elementary integral calculations and the ODE comparison principle. We now restate these results as follows.
Lemma 2.2. Assume that the assumptions given in Lemma 2.1 are satisfied. Then,
Moreover, for any ,
where .
Next, we present some well-known estimates for the diffusion-semigroup subject to homogeneous Neumann boundary conditions; e.g., see [1, Lemma 3.5]. For further details, see [38,39]. For , the operator , defined on a dense domain, is the closed fractional powers of the sectorial operator in for .
Lemma 2.3. Let , , , and be constants.
(i) If
then there exists a constant such that for all ,
(ii) If and , then there are two constants and such that for all , the so-called estimate
holds, where and the associated heat semigroup maps into .
Now, using the two preceding lemmas, we additionally derive a priori estimates for solutions of (1.1).
Lemma 2.4. Assume that the assumptions in Lemma 2.1 hold and that . Then, for any , there exists a positive constant such that
Proof. We begin by choosing an . Note that all constants and introduced below are fixed and will be used without further specification.
From the representation formulation of the first equation in system (1.1), one can obtain that for ,
(2.1) |
where
We choose a
Applying Lemma 2.3(ⅰ) to (2.1) and then using the triangle inequality, one has that
(2.2) |
According to Lemma 2.3(ⅱ) and the assumption on , we obtain that
(2.3) |
Moreover, by using assumptions (H1) and (H2) and applying Lemmas 2.2 and 2.3(ⅱ), we deduce that
(2.4) |
Here, the last inequality follows from elementary integral calculus due to the fact that , and is independent of but depends on . Thus, by applying (2.3) and (2.4) in (2.2), we see that for all ,
Combining this result with the assumption on for sufficiently small , we obtain the desired conclusion, thereby completing the proof.
Lemma 2.5. Assume that the assumptions in Lemma 2.1 hold and that . Then, for each , there exists a positive constant such that
Proof. Multiplying the second equation of system (1.1) by and integrating the resulting equation on , and then applying integration by parts, we can derive
(2.5) |
By inserting the simple facts that
into (2.5), we can immediately obtain that
(2.6) |
Furthermore, applying assumptions (H2)–(H4) to (2.6), we deduce that the right-hand side of (2.6) is bounded above as follows:
(2.7) |
By setting
and sequentially applying Hölder's inequality with exponents and , Lemma 2.4, and Hölder's inequality with exponents and , it is easy to see that the following holds:
(2.8) |
where is a positive constant depending on . After choosing an to satisfy
and incorporating the previously obtained inequality into (2.7), followed by applying Young's inequality, we obtain that
Since , it is clear that . Thus, we also note that Young's inequality with exponents and implies that
for a fixed constant chosen to satisfy
This leads to the following inequality:
(2.9) |
Finally, applying Hölder's inequality with exponents and yields the inequality
Substituting this into (2.9) and setting , we eventually obtain that
(2.10) |
for all . Using the fact that , we apply an ODE comparison argument to deduce from (2.10) that there exists a positive constant such that for all . Thus, the proof is complete.
When , inequality (2.8) cannot be derived. In fact, Lemma 2.4 is no longer applicable in this case. This highlights the essential role of the one-dimensional setting () in our analysis.
Using Lemma 2.5, we obtain the following improved result compared to Lemma 2.4.
Lemma 2.6. Assume that the assumptions given in Lemma 2.1 hold and that . Then, there exists a positive constant such that
Proof. We first choose an . As demonstrated in the proof of Lemma 2.4, we will use the constants and mentioned below without explicit reference; all of these constants are positive and independent of .
For fixed
applying Lemma 2.3(ⅰ) to (2.1), we derive that
As in (2.3), we derive that holds for all . Additionally, by applying assumptions (H1) and (H2) along with Lemmas 2.2, 2.3(ii), and 2.5, we obtain the following result analogous to (2.4):
Therefore, as in the final part of the proof of Lemma 2.4, we conclude that our claim is valid.
We now need to establish the -estimate of , for which the following Gagliardo-Ladyzhenskaya-Nirenberg inequality (see [42, p. 63]) is required.
Lemma 2.7. Let be a bounded domain in with a smooth boundary. Then, for all , there exists a constant such that
where .
Lemma 2.8. Assume that the assumptions stated in Lemma 2.1 are satisfied and that . Then, there exists a positive constant such that
Proof. We begin with the following inequality, which is obtained directly by applying assumptions (H2) and (H4) to (2.6) for :
(2.11) |
The remaining steps follow from standard Moser-type iteration arguments (see, e.g., [4,26,35,40]). For completeness, the full proof is provided in the Appendix.
We are now finally prepared to prove our main result, Theorem 1.1.
Proof of Theorem 1.1. First, we note from Lemma 2.2 that for all . From Lemma 2.1, we see that system (1.1) possesses a classical and local-in-time solution on . Hence, together with Lemmas 2.6 and 2.8, the extensibility-criterion presented in Lemma 2.1 enables us to conclude . This completes the proof.
We now consider the bifurcation of nonconstant positive steady-state solutions of system (1.2). To begin, we derive a priori estimates for the positive solutions of (1.2).
Lemma 2.9. Assume that (H1), (H2), (H3*), and (H4) hold. Let be a positive solution of (1.2). Then there exists a positive constant such that
Proof. By directly applying the maximum principle to the first equation in (1.2), we can immediately see that .
From the result obtained by integrating the equations in (1.2) over , we can readily derive the following:
which, together with (H1) and (H3*), implies that
(2.12) |
For any , multiplying the second equation of system (1.2) by and integrating the resulting equation on , and then applying integration by parts, we can derive
(2.13) |
By inserting the simple fact that
into (2.13) and using assumptions (H2) and (H4), we can obtain that
(2.14) |
Similarly, by multiplying the first equation of system (1.2) by , integrating the resulting equation over , and applying integration by parts, we obtain the following:
(2.15) |
Then, using (2.14), (2.15), assumptions (H2) and (H4), and the result obtained by applying our first finding to satisfying (H1), we arrive at the following:
(2.16) |
Applying Hölder's inequality with exponents and , and then using (2.12), we can deduce that
(2.17) |
where is a positive constant independent of . In particular, for , Hölder's inequality gives that
(2.18) |
Moreover, since , the Sobolev inequality implies that for any ,
(2.19) |
where is a positive constant independent of . We now apply (2.17)–(2.19), which together yield
(2.20) |
where
Using (2.20) with such that for , we obtain
If we set , then the above becomes
(2.21) |
Similarly, using (2.20) with such that for , we obtain
If we set , then the above becomes
(2.22) |
By iteratively using the inequalities in (2.21) and (2.22), we are finally led to the following:
(2.23) |
where
for a positive constant determined by the definition of .
Taking the limit as in (2.23) gives
Consequently, by using (2.12) together with the following inequality to this estimate,
which is obtained via Hölder's inequality, we can complete this proof.
By combining the above result that all positive solutions of (1.2) are bounded in with the -theory of elliptic equations, the Schauder estimates, and the Sobolev embedding theorem, we can conclude that , for some (e.g., see [9,31]).
With respect to (1.2), we will obtain a nonconstant positive solution branch that bifurcates from the positive constant equilibrium (if it exists), under the assumptions of Theorem 1.3. Let be a bifurcation parameter and assume that all other coefficients are fixed. For , we define Banach spaces and as
Furthermore, we set a mapping by
where and were defined before Theorem 1.3. It is obvious that is continuously differentiable in an open set in , and that for all , where is a positive constant steady state of (1.2). It follows that the Fréchet derivative of at is given by
where
In particular, when (provided that the positive constant solution exists), a straightforward computation yields
(2.24) |
We now identify potential bifurcation values of . To find possible bifurcation points corresponding to values of , we consider the linearized problem and examine whether it admits a nontrivial solution. Using the eigenfunction expansions and in this problem, we can derive the following:
We see that the linearized problem has nontrivial solutions if and only if
(2.25) |
that is to say, for , where was defined in Theorem 1.3. Here, the case can be readily excluded, since assumptions (H4) and (H5) imply that
A detailed verification of this computation can be found in the proof of [1, Theorem 2.7].
Lemma 2.10. Suppose that all assumptions of Theorem 1.3 hold. Then .
Proof. Suppose . Then as in the previous discussions, using the eigenfunction expansions of and , we see that , since the definition of gives (2.25). Thus, a direct calculation shows that
(2.26) |
Moreover, we observe that
Thus, the assumption given in Theorem 1.3 ensures that for . Hence, the assertion holds, as is a simple eigenvalue.
Lemma 2.11. Suppose that all assumptions of Theorem 1.3 hold. Then .
Proof. Consider the adjoint operator of :
Following the same method as in the proof of Lemma 2.10, we obtain
(2.27) |
Suppose . Then there exists such that
where is a complement of in . We easily see that
Thus,
(2.28) |
gives .
Lemma 2.12. Suppose that all assumptions of Theorem 1.3 hold. Then , where .
Proof. Suppose that . Then, it is obvious from (2.26)–(2.28) that
and
which is a contradiction.
We are now finally prepared to prove our main result, Theorem 1.3.
Proof of Theorem 1.3. By virtue of Lemmas 2.10–2.12, the first result follows from the local bifurcation theorem [41].
For the global bifurcation analysis, we consider the interval . In view of the implicit function theorem, the linearization around the trivial solution and the semi-trial solution of system (1.2) shows that any solutions bifurcating from or are not positive near the bifurcation points. Thus, the positive solution branches cannot be joined to or . According to the global bifurcation theory [36], one of the following two alternatives must be satisfied:
(ⅰ) is unbounded in ;
(ⅱ) meets a point with .
We know from Lemma 2.9 that any positive solution to (1.2) is bounded in . Furthermore, from the discussion presented below Lemma 2.9, is also bounded in . Therefore, the global branch is bounded in , which implies that must be unbounded in the -direction; that is, joins to in .
In this paper, building upon the considerations outlined above and motivated by Remark 2.2 in [1], we established the global existence and boundedness of classical solutions to system (1.1) in the one-dimensional case (i.e., ) for arbitrary positive prey-taxis coefficients . To achieve this result, we introduced a stronger logistic assumption on the predator growth term, denoted as (H3*). This assumption plays a central role in providing sufficient control over population densities, thereby preventing population overcrowding without imposing the restrictive truncation or smallness conditions previously required.
Our analysis relied on deriving a priori estimates through a careful combination of the Moser-type iteration method, weighted integral estimates, and precise applications of the Gagliardo-Ladyzhenskaya-Nirenberg inequality. In particular, the logistic source introduced by condition (H3*), especially through the exponent , was essential in ensuring the boundedness of all positive steady states for arbitrary values of . This boundedness property played a crucial role in enabling the application of global bifurcation theory in the steady-state analysis.
In the latter part of the paper, we applied global bifurcation theory to establish the existence of nonconstant positive steady states, thereby providing a rigorous basis for pattern formation in the model. Such spatial patterns, induced by prey-taxis and cooperative hunting effects, offer insight into how local interactions and movement mechanisms can generate heterogeneous population distributions even in homogeneous environments.
Our results extend and complement the existing literature, particularly the discussions in Remark 2.2 of [1], and confirm the conditions under which population densities remain controlled in one-dimensional predator-prey models with prey-taxis. For future research, an important and challenging direction will be to explore whether similar global existence and boundedness results can be achieved in higher-dimensional scenarios (i.e., ) or in the absence of logistic source terms, thereby broadening the class of predator-prey models for which global existence can be established.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The authors would like to express their sincere appreciation to the associate editor and the anonymous reviewers for their valuable suggestions, which encouraged the addition of bifurcation analysis and contributed to enhancing the manuscript. The first author (K. Ryu) was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2021R1A2C1095325).
The authors declare there are no conflicts of interest.
[1] |
W. Ko, K. Ryu, A diffusive predator-prey system with hunting cooperation in predators and prey-taxis: Ⅰ global existence and stability, J. Math. Anal. Appl., 543 (2025), 129005. https://doi.org/10.1016/j.jmaa.2024.129005 doi: 10.1016/j.jmaa.2024.129005
![]() |
[2] |
B. Griffith, Group predator defense by mule deer in Oregon, J. Mammal., 69 (1988), 627–629. https://doi.org/10.2307/1381359 doi: 10.2307/1381359
![]() |
[3] |
Y. Sait, Prey kills predator: counter-attack success of a spider mite against its specific phytoseiid predator, Exper. Appl. Acarol., 2 (1986), 47–62. https://doi.org/10.1007/BF01193354 doi: 10.1007/BF01193354
![]() |
[4] |
E. C. Haskell, J. Bell, Pattern formation in a predator-mediated coexistence model with prey-taxis, Discrete Contin. Dyn. Syst. - Ser. B, 25 (2020), 2895–2921. https://doi.org/10.3934/dcdsb.2020045 doi: 10.3934/dcdsb.2020045
![]() |
[5] |
J. M. Lee, T. Hillen, M. A. Lewis, Continuous traveling waves for prey-taxis, Bull. Math. Biol., 70 (2008), 654–676. https://doi.org/10.1007/s11538-007-9271-4 doi: 10.1007/s11538-007-9271-4
![]() |
[6] |
R. Tyson, S. R. Lubkin, J. D. Murray, Model and analysis of chemotactic bacterial patterns in a liquid medium, J. Math. Biol., 38 (1999), 359–375. https://doi.org/10.1007/s002850050153 doi: 10.1007/s002850050153
![]() |
[7] |
C. Cosner, D. L. DeAngelis, J. S. Ault, D. B. Olson, Effects of spatial grouping on the functional response of predators, Theor. Popul. Biol., 56 (1999), 65–75. https://doi.org/10.1006/tpbi.1999.1414 doi: 10.1006/tpbi.1999.1414
![]() |
[8] |
L. Berec, Impacts of foraging facilitation among predators on predator-prey dynamics, Bull. Math. Biol., 72 (2010), 94–121. https://doi.org/10.1007/s11538-009-9439-1 doi: 10.1007/s11538-009-9439-1
![]() |
[9] |
W. Ko, K. Ryu, A diffusive predator-prey system with hunting cooperation in predators and prey-taxis: Ⅱ stationary pattern formation, J. Math. Anal. Appl., 543 (2025), 128947. https://doi.org/10.1016/j.jmaa.2024.128947 doi: 10.1016/j.jmaa.2024.128947
![]() |
[10] |
K. Ryu, W. Ko, On dynamics and stationary pattern formations of a diffusive predator-prey system with hunting cooperation, Discrete Contin. Dyn. Syst. - Ser. B, 27 (2022), 6679–6709. https://doi.org/10.3934/dcdsb.2022015 doi: 10.3934/dcdsb.2022015
![]() |
[11] |
K. Ryu, W. Ko, M. Haque, Bifurcation analysis in a predator-prey system with a functional response increasing in both predator and prey densities, Nonlinear Dyn., 94 (2018), 1639–1656. https://doi.org/10.1007/s11071-018-4446-0 doi: 10.1007/s11071-018-4446-0
![]() |
[12] |
D. Sen, S. Ghorai, M. Banerjee, Allee effect in prey versus hunting cooperation on predator - enhancement of stable coexistence, Int. J. Bifurcation Chaos, 29 (2019), 1950081. https://doi.org/10.1142/S0218127419500810 doi: 10.1142/S0218127419500810
![]() |
[13] |
K. Vishwakarma, M. Sen, Influence of Allee effect in prey and hunting cooperation in predator with Holling type-Ⅲ functional response, J. Appl. Math. Comput., 68 (2022), 249–269. https://doi.org/10.1007/s12190-021-01520-1 doi: 10.1007/s12190-021-01520-1
![]() |
[14] |
M. T. Alves, F. M. Hilker, Hunting cooperation and Allee effects in predators, J. Theor. Biol., 419 (2017), 13–22. https://doi.org/10.1016/j.jtbi.2017.02.002 doi: 10.1016/j.jtbi.2017.02.002
![]() |
[15] |
H. Amann, Dynamic theory of quasilinear parabolic equtions. Ⅱ. Reaction-diffusion systems, Differ. Integr. Equations, 3 (1990), 13–75. https://doi.org/10.57262/die/1371586185 doi: 10.57262/die/1371586185
![]() |
[16] | H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in Function Spaces, Differential Operators and Nonlinear Analysis (eds. H. Schmeisser and H. Triebel), Springer Fachmedien Wiesbaden, (1993), 9–126. https://doi.org/10.1007/978-3-663-11336-2_1 |
[17] | A. Friedman, Partial Differential Equations, Dover, New York, 2008. |
[18] |
H. Jin, Z. Wang, Global stability of prey-taxis systems, J. Differ. Equations, 262 (2017), 1257–1290. https://doi.org/10.1016/j.jde.2016.10.010 doi: 10.1016/j.jde.2016.10.010
![]() |
[19] |
Y. Li, J. Lankeit, Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion, Nonlinearity, 29 (2016), 1564–1595. https://doi.org/10.1088/0951-7715/29/5/1564 doi: 10.1088/0951-7715/29/5/1564
![]() |
[20] |
M. Winkler, On the Cauchy problem for a degenerate parabolic equation, Z. Anal. Anwend., 20 (2001), 677–690. https://doi.org/10.4171/ZAA/1038 doi: 10.4171/ZAA/1038
![]() |
[21] |
B. E. Ainseba, M. Bendahmane, A. Noussair, A reaction-diffusion system modeling predator-prey with prey-taxis, Nonlinear Anal. Real World Appl., 9 (2008), 2086–2105. https://doi.org/10.1016/j.nonrwa.2007.06.017 doi: 10.1016/j.nonrwa.2007.06.017
![]() |
[22] |
T. Hillen, K. Painter, A users guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183–217. https://doi.org/10.1007/s00285-008-0201-3 doi: 10.1007/s00285-008-0201-3
![]() |
[23] |
Y. Tao, Global existence of classical solutions to a predator-prey model with nonlinear prey-taxis, Nonlinear Anal. Real World Appl., 11 (2010), 2056–2064. https://doi.org/10.1016/j.nonrwa.2009.05.005 doi: 10.1016/j.nonrwa.2009.05.005
![]() |
[24] |
S. Wu, J. Shi, B. Wu, Global existence of solutions and uniform persistence of a diffusive predator-prey model with prey-taxis, J. Differ. Equations, 260 (2016), 5847–5874. https://doi.org/10.1016/j.jde.2015.12.024 doi: 10.1016/j.jde.2015.12.024
![]() |
[25] |
Y. Cai, Q. Cao, Z. A. Wang, Asymptotic dynamics and spatial patterns of a ratio-dependent predator-prey system with prey-taxis, Applicalbe Anal., 101 (2022), 81–99. https://doi.org/10.1080/00036811.2020.1728259 doi: 10.1080/00036811.2020.1728259
![]() |
[26] |
Q. Wang, C. Gai, J. Yan, Qualitative analysis of a Lotka-Volterra competition system with advection, Discrete Contin. Dyn. Syst., 35 (2015), 1239–1284. https://doi.org/10.3934/dcds.2015.35.1239 doi: 10.3934/dcds.2015.35.1239
![]() |
[27] |
Y. Tao, M. Winkler, Large time behavior in a forager-exploiter model with different taxis strategies for two groups in search of food, Math. Models Methods Appl. Sci., 29 (2019), 2151–2182. https://doi.org/10.1142/S021820251950043X doi: 10.1142/S021820251950043X
![]() |
[28] |
J. Wang, Global existence and stabilization in a forager-exploiter model with general logistic sources, Nonlinear Anal. Theory Methods Appl., 222 (2022), 112985. https://doi.org/10.1016/j.na.2022.112985 doi: 10.1016/j.na.2022.112985
![]() |
[29] |
C. Wang, Z. Zheng, The effects of cross-diffusion and logistic source on the boundedness of solutions to a pursuit-evasion model, Electron. Res. Arch., 31 (2023), 3362–3380. https://doi.org/10.3934/era.2023170 doi: 10.3934/era.2023170
![]() |
[30] |
M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Commun. Partial Differ. Equations, 37 (2012), 319–351. https://doi.org/10.1080/03605302.2011.591865 doi: 10.1080/03605302.2011.591865
![]() |
[31] |
L. Dung, H. L. Smith, Steady states of models of microbial growth and competition with chemotaxis, J. Math. Anal. Appl., 229 (1999), 295–318. https://doi.org/10.1006/jmaa.1998.6167 doi: 10.1006/jmaa.1998.6167
![]() |
[32] |
D. Luo, Q. Wang, Global bifurcation and pattern formation for a reaction-diffusion predator-prey model with prey-taxis and double Beddington-DeAngelis functional responses, Nonlinear Anal. Real World Appl., 67 (2022), 103638. https://doi.org/10.1016/j.nonrwa.2022.103638 doi: 10.1016/j.nonrwa.2022.103638
![]() |
[33] |
X. Wang, W. Wang, G. Zhang, Global bifurcation of solutions for a predator-prey model with prey-taxis, Math. Methods Appl. Sci., 38 (2015), 431–443. https://doi.org/10.1002/mma.3079 doi: 10.1002/mma.3079
![]() |
[34] |
S. Wu, J. Wang, J. Shi, Dynamics and pattern formation of a diffusive predator-prey model with predator-taxis, Math. Models Methods Appl. Sci., 28 (2018), 2275–2312. https://doi.org/10.1142/S0218202518400158 doi: 10.1142/S0218202518400158
![]() |
[35] |
N. Alikakos, bounds of solutions of reaction-diffusion equations, Commun. Partial Differ. Equations, 4 (1979), 827–868. https://doi.org/10.1080/03605307908820113 doi: 10.1080/03605307908820113
![]() |
[36] |
R. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487–513. https://doi.org/10.1016/0022-1236(71)90030-9 doi: 10.1016/0022-1236(71)90030-9
![]() |
[37] |
Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521–529. https://doi.org/10.1016/j.jmaa.2011.02.041 doi: 10.1016/j.jmaa.2011.02.041
![]() |
[38] |
D. Horstmann, M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equations, 215 (2005), 52–107. https://doi.org/10.1016/j.jde.2004.10.022 doi: 10.1016/j.jde.2004.10.022
![]() |
[39] |
M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr., 283 (2010), 1664–1673. https://doi.org/10.1002/mana.200810838 doi: 10.1002/mana.200810838
![]() |
[40] |
Y. Luo, Global existence and stability of the classical solution to a density-dependent prey-predator model with indirect prey-taxis, Math. Biosci. Eng., 18 (2021), 6672–6699. https://doi.org/10.3934/mbe.2021331 doi: 10.3934/mbe.2021331
![]() |
[41] |
M. G. Crandall, P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321–340. https://doi.org/10.1016/0022-1236(71)90015-2 doi: 10.1016/0022-1236(71)90015-2
![]() |
[42] | O. Ladyzhenskaya, V. Solonnikov, N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, RI, 1968. https://doi.org/10.1090/mmono/023 |