1. Introduction
Since Lotka [
1] and Volterra [
2] established the classic predator–prey model, the study of predator–prey systems has become a central research topic in mathematical ecology. Due to the universality and ecological significance of predator–prey interactions in nature, this dynamic system has attracted the continuous attention of mathematical and ecological researchers and has become the main focus of ecological and mathematical ecology research [
3,
4,
5,
6]. As one of the basic models of predator–prey systems, the food chain model has been widely studied by scholars. The study of the food chain model has advanced the understanding of ecological interactions from linear relationships to complex nonlinear systems.
The three-species food chain model provides a theoretical framework for explaining the complexity of ecosystems by simulating multi-trophic-level interactions. Its dynamic behavior not only reflects the spatiotemporal transmission characteristics of trophic cascades, but also clarifies the key role of nonlinear mechanisms in controlling energy flow and stability regulation between levels [
7,
8,
9,
10,
11,
12,
13]. In 1991, Hastings and Powell [
14] developed a three-species food chain model within the classic Lotka–Volterra framework:
where population
denotes the prey, population
represents the middle predator, and population
corresponds to the top predator. The parameters
and
represent the predation efficiency of population
on population
and population
on population
, respectively. The coefficients
and
denote the saturation effects of population
density on the predation efficiency of population
and population
density on the predation efficiency of population
, respectively. Additionally,
and
correspond to the natural mortality rates of populations
and
. The study revealed that when mortality rates
and
of intermediate predator
and top predator
reach critical thresholds, the system transitions from a stable equilibrium to chaotic oscillations [
14]. This groundbreaking finding not only confirms the universality of deterministic chaos in ecosystems but also advances further exploration into food chain models.
It is well known that both past historical states and current conditions significantly affect population dynamics. In biological systems, time delays are inevitable due to factors such as maturation time, gestation period, and digestion time. Introducing time delays in food chain models can not only enhance the realism of these models, but also increase their complexity. Time delay systems retain all the dynamic characteristics of non-delayed systems while exhibiting richer behavioral patterns. Crucially, the addition of delays can disrupt the originally stable equilibrium state and lead to population fluctuations. Therefore, studying the dynamics of time-delayed food chain models has dual significance—mathematical and biological [
15,
16,
17,
18].
Traditional food chain models often include maturation delays and gestation delays [
19,
20,
21,
22,
23]. Tian, B. et al. introduced digestion delay (also referred to as energy transfer delay) into the food chain model [
24]. It offers diverse perspectives for advancing the dynamic analysis of food chain models. In natural ecosystems, energy transfer across trophic levels inherently involves hysteresis effects [
25]. For example, predators take days to weeks to digest prey (e.g., pythons), while juveniles take months or even years to mature into adults (e.g., salmon). These delays directly regulate population fluctuation patterns. Therefore, introducing energy transfer delays in food chain models can not only reveal the evolutionary dynamics of predator–prey systems more realistically, but also significantly improve simulation accuracy and practical guidance value for ecosystem management.
Food chain models incorporating energy transfer time lags are of considerable research significance. Specifically, the introduction of dual delays in energy transfer—from prey to mesopredator and from mesopredator to apex predator—within tri-trophic systems allows for a more refined characterization of cascading response mechanisms in real-world ecosystems. These dual delays enable a more accurate depiction of the cascading response mechanisms characteristic of real ecosystems. Building upon system (1), this paper extends the framework by introducing a top predator, thereby constructing a three-species food chain model. We incorporate two essential time delays: (i) the delay from prey consumption to its conversion into predator biomass, and (ii) the delay involved in transferring biomass from intermediate predators to top predators. The resulting three-species food chain model is expressed as follows:
By analyzing the coupling of energy transfer delays across multiple trophic levels, this work explores how synergistic delays destabilize equilibrium states, leading to Hopf bifurcations and emergent oscillatory regimes. System (2) incorporates both intraspecific competition within populations and the effects of predation behavior and time delays on population dynamics. In this system, population
denotes the prey, population
represents the middle predator, and population
corresponds to the top predator. The variables
and
represent the population densities of the three species, respectively,
is the natural growth rate of population
,
denotes the intraspecific competition coefficient of population
, and
and
are the interspecific interaction coefficients between population
and population
, and between population
and population
, respectively.
and
represent the intrinsic growth rates of predator populations
and
, while
and
correspond to their resource competition coefficients. The time delays
and
describe the energy conversion lags from population
to population
and from population
to population
, respectively. All parameters,
,
,
,
,
and
, are positive constants. Based on different values of the two delays,
and
, this paper establishes stability conditions for the internal equilibrium of system (2) and derives criteria for the existence of Hopf bifurcations under five distinct scenarios. The introduction of dual delays significantly enhances the modeling depth of ecological dynamics. Compared to single-delay systems [
26], this dual-delay framework more realistically reconstructs the cascading response mechanisms across trophic levels by simultaneously capturing the lag effects in energy transfer between different hierarchical layers. This dual-delay coupling significantly alters the population dynamic equilibria. For example, in the Northwest Atlantic cod ecosystem, seasonal fluctuations in plankton abundance propagate through two trophic levels, resulting in a 4–6-month lag in cod population peaks relative to the initial plankton signal. This temporal mismatch critically impacts the timing and sustainability of fishery harvest cycles. Furthermore, by applying normal form theory and the center manifold theorem, we derive explicit formulae to determine the direction of Hopf bifurcations and the stability of periodic solutions in system (2). Finally, numerical simulations are presented to validate the theoretical results.
2. Stability of Equilibrium and Existence of Hopf Bifurcation
Let
, the transformed variables,
, are still denoted as
and
remains as
; then, system (2) is transformed into
where
.
By setting the right-hand-side functions of system (3) to zero, the internal equilibrium,
, can be obtained:
Clearly, when the condition holds, system (3) admits an internal equilibrium.
The Jacobian matrix of system (3) at the equilibrium
is given by
where
The characteristic equation of system (3) at
is
where
The stability of system (3) at and the existence of Hopf bifurcation are analyzed under five distinct cases.
For clarity, we introduce the hypothesis
Based on the Routh–Hurwitz criterion, the following theorem is established.
Theorem 1. If and hold, then the internal equilibrium of system (3) is asymptotically stable.
Let
be a root of Equation (5). Substituting
into Equation (5) and separating the real and imaginary parts, we obtain
Squaring and adding both equations in (6) yields
where
Lemma 1. If or , hold, then Equation (7) admits a positive root, where , and are demonstrated in the proof of Lemma 1.
Proof. Let . Substituting this into Equation (7), we obtain
For clarity, let us define
Obviously, .
When , Equation (8) has at least one positive root.
When
, taking the derivative of Equation (11), we obtain
. When
, we can imply that
and
is monotonically increasing. Therefore, when
, Equation (10) has no positive roots; when
,
has two real roots
Therefore, when and , Equation (8) admits positive roots. This completes the proof. □
From positive roots
and
of Equation (8), the roots of Equation (7) can be derived as
. For each fixed
, there exists a corresponding sequence
, where
When , becomes a pair of pure imaginary roots of Equation (5).
Let and be the corresponding . Suppose is a root of Equation (5) at , satisfying and .
Lemma 2. If holds, then .
Proof. Differentiating both sides of Equation (5) with respect to , we obtain
Substituting
into (10) and extracting the real part yields
Therefore, when the hypothesis holds, we have . This completes the proof. □
From Lemmas 1 and 2, the following theorem is derived.
Theorem 2. If are satisfied, then when , the internal equilibrium of system (3) is asymptotically stable. When , the internal equilibrium of system (3) is unstable. When , system (3) exhibits a Hopf bifurcation at the internal equilibrium .
Let
be a root of Equation (11). Substituting
into Equation (11) and separating the real and imaginary parts, we obtain
Squaring and adding both equations in Equation (12) yields
where
Lemma 3. If or , holds, then Equation (13) admits positive roots, where , and are demonstrated in the proof of Lemma 3.
Proof. Let . Substituting this into Equation (13), we obtain
Obviously, .
When , Equation (13) has at least one positive root.
When
, taking the derivative of Equation (15), we obtain
. When
, we can imply that
, and
is monotonically increasing. Therefore, when
, Equation(13) has no positive roots. When
,
has two real roots
Therefore, when , , Equation (14) admits positive roots. This completes the proof. □
From positive roots
and
of Equation (14), the roots of Equation (13) can be derived as
. For each fixed
, there exists a corresponding sequence,
, where
When , becomes a pair of pure imaginary roots of Equation (11).
Let and be the corresponding . Suppose is a root of Equation (11) at , satisfying and .
Lemma 4. If holds, then .
Proof. Differentiating both sides of Equation (11) with respect to , we obtain
Substituting
into Equation (16) and extracting the real part, we derive
Therefore, when the hypothesis holds, we have . This completes the proof. □
From Lemmas 3 and 4, the following theorem is derived.
Theorem 3. If are satisfied, then when , the internal equilibrium of system (3) is asymptotically stable. When , the internal equilibrium of system (3) is unstable. When , system (3) exhibits a Hopf bifurcation at the internal equilibrium .
Case 4. , where is constrained within its stability interval, while is treated as a bifurcation parameter.
The characteristic equation of system (3) at the equilibrium
is
Let
be a root of Equation (17). Substituting
into Equation (17) and separating the real and imaginary parts, we obtain
Squaring and adding both equations in Equation (18) yields
where
Lemma 5. If holds, then Equation (19) admits at least one positive root.
Proof. Define the function
Since
, when
, Equation (19) possesses at least one positive root. Due to the sixth-degree polynomial nature of
, Equation (19) has finitely many positive real roots, denoted as
. For each fixed
, there exists a corresponding sequence,
, where
When , becomes a pair of pure imaginary roots of Equation (17).
Let , and be the corresponding . Suppose is a root of Equation (17) at the critical delay , satisfying and . □
Lemma 6. If holds, then , where , , and are demonstrated in the proof of Lemma 6.
Proof. Differentiating both sides of Equation (17) with respect to , we obtain
Substituting
into Equation (20) and extracting the real part, we derive
where
Therefore, when the hypothesis holds, we have . This completes the proof. □
From Lemmas 5 and 6, the following theorem is derived.
Theorem 4. If are satisfied, then when , the internal equilibrium of system (3) is asymptotically stable. When , the internal equilibrium of system (3) is unstable. When , system (3) exhibits a Hopf bifurcation at the internal equilibrium .
Case 5. , where is constrained within its stability interval, while is treated as a bifurcation parameter.
The characteristic equation of system (3) at the equilibrium
is
Let
be a root of Equation (21). Substituting
into Equation (21) and separating the real and imaginary parts, we obtain
Squaring and adding both equations in Equation (22) yields
where
Lemma 7. If holds, then Equation (23) admits at least one positive root.
Proof. Define the function
Since
, when
, Equation (23) possesses at least one positive root. Due to the sixth-degree polynomial nature of
, Equation (23) has finitely many positive real roots, denoted as
. For each fixed
, there exists a corresponding sequence
, where
When , becomes a pair of pure imaginary roots of Equation (21).
Let , and be the corresponding . Suppose is a root of Equation (21) at the critical delay , satisfying and . □
Lemma 8. If holds, then , where , , and are demonstrated in the proof of Lemma 8.
Proof. Differentiating both sides of Equation (21) with respect to , we obtain
Substituting
into Equation (24) and extracting the real part, we derive
where
Therefore, when the hypothesis holds, we have . This completes the proof. □
From Lemmas 7 and 8, the following theorem is derived.
Theorem 5. If are satisfied, then when , the internal equilibrium of system (3) is asymptotically stable. When , the internal equilibrium of system (3) is unstable. When , system (3) exhibits a Hopf bifurcation at the internal equilibrium .