Funding: This work is supported by the National Natural Science Foundation of China (no. 12271308), the China Scholarship Council (no. 202308370307), and the SDUST Innovation Fund for Graduate Students (YC20210231).
ABSTRACT
The authors propose a new predator–prey game model by integrating two optional strategies into prey species: cooperative and isolation strategies. An investigation of the evolutionary impact on predator–prey system dynamics is given. The model utilizes a replicator equation to track changes in the frequency of cooperative strategy among preys, along with two population growth equations detailing variations in prey and predator biomass. Interestingly, both cooperative and isolated prey strategies can coexist within the prey species. As the predation rate increases, preys are more inclined to adopt cooperative strategies, thereby boosting predator biomass. Abundant resources then encourage preys to opt for cooperative strategies to secure resources and defend against predators. Under specific conditions, the system exhibits intricate dynamics involving bistability, transcritical, and Hopf bifurcations. A novel fractional-order predator–prey game model is then derived using a continuous-time random walk based on a potential physical stochastic process, which is different than the traditional fractional-order models. The influence of fractal dimension on system dynamics is analyzed, showing that it can either diminish or eliminate cooperative strategies within prey species and potentially lead to predator extinction. Interestingly, dimension affects the convergence speed of the system without altering its final steady state: As the fractal dimension and time change, the frequency of cooperative strategies adopted by preys, the biomasses of prey, and predator species ultimately stabilize toward the same steady state, but their rates of approaching to the steady state are different. In this case, only preys survive and cooperative and isolated strategies coexist in prey species. These models can have multiple applications in different settings, aiding with management of animal populations, pest control, tumor immune interactions, and infectious diseases.
1 Introduction
In ecological models, the predator–prey paradigm [1-7] has a long and outstanding history: It has been studied by many researchers and has been found to exhibit rich dynamic behaviors. The researchers also explored the effects of factors on the predator–prey system, such as fear [8], prey refuge [9], mutation interference [10, 11], and Allee effect [14, 12, 13], among others. However, the behavioral structure of individuals in predator–prey models has not been thoroughly considered, in particular with regards to cooperation and isolation. For instance, when the prey faces resources, some individuals adopt a cooperative strategy and collectively act to obtain resources before sharing them. When facing predators, cooperative prey engages in group defense [18, 15-17], while isolated preys enjoy resources alone and resist the capture of predators alone. In a work by Sasmal et al. [16], a predator–prey model with fear and group defense is considered. Similarly, in this work, we incorporate the behavior strategy of the prey into the predator–prey framework. Although there are currently many research articles on predator–prey dynamics, there is limited research on how the dynamics of prey strategy behavior affect the overall dynamics of ecological models.
In this regard, evolutionary game theory [19-23] provides a powerful framework for studying strategy changes, strategy stability, and the dynamics of individual interactions with different strategies. By utilizing the replicator equation [24-26], the game model displays the frequency change of individuals adopting various strategies in the population. This is reasonably new, as there are few examples that simultaneously incorporate individual behavior dynamics using game theory methods and population dynamics. For example, it is worth noting that Auger et al. [26] established a predator–prey model with Hawk and Dove strategies, where they explored the dynamic effects of individual predator behavior on the predator–prey system. Aggressive behavior is related to high prey biomass and low predator biomass, while Hawk and Dove strategies coexist at low prey biomass and high predator biomass. Galanthay et al. [27] proposed instead a new consumer-resource model, in which consumers have two strategies: Hawk and Dove, similarly. They explored the dynamic behavior of the model and made new predictions on how the degree of attack varies with resource richness, species mortality, and game duration. Sahoo et al. [28] considered a snowdrift game in predator species to explore the evolution of cooperation, and studied the dynamical impact of strategies on predator–prey interactions. Their analysis showed that the prey population becomes polymorphic, with cooperative and defective prey coexisting. Meanwhile, they also found that a lower cooperation frequency would lead to the extinction of prey and predator, while a higher cooperation frequency of prey would lead to the extinction of predators. There are also some other important studies on how the behavioral dynamics of a species affect the qualitative behavior of ecological models [29-31]. In this article, we first consider that preys have different strategies to absorb resources and resist predator capture, with different strategies corresponding to different benefits and costs. We combine fast time game dynamics and slow time population growth dynamics to explore the effects of cooperative and isolated strategies on the biomass of prey and predators, with new insights.
In some populations, the dynamics of the system depend not only on the current state but also on the historical state of the system. Many researchers thus have extended the classical coupled ODEs by using fractional-order time derivatives [32-36], incorporating historical dependencies into dynamics. This generalization is typically achieved by replacing the integer-order time derivative with a fractional-order Caputo derivative, defined such that the derivative of a function depends on the entire history of that function. These fractional models are mathematically interesting but may not match potential physical processes. To address these issues, Angstmann et al. [37-40] developed a comprehensive framework for formulating fractional-order epidemic models, using generalized continuous-time random walk (CTRW) with waiting times that moderate transitions between compartments. In this article, we introduce this method into the predator–prey system with a prey game, considering how the capture of prey by predators in the past affects the current predator biomass.
The purpose of this work is to first explore the impact of individual behavior of prey on the biomass of prey and predators and analyze different strategies adopted by preys on the survival and extinction of predators. Furthermore, the impact of past predator capture prey on the current predator biomass changes is studied. We consider two different time scales, with the fast time scale corresponding to the selection of cooperative and isolated strategies by the prey and the slow time scale corresponding to the growth of the prey, natural death of the predator, and interaction between the prey and the predator. We combine fast time game dynamics and slow population growth dynamics and derive a new fractional-order predator–prey game model through a potential physical stochastic process with CTRW.
The structure of this article is as follows: In Section 2, we firstly combine the fast game dynamics and the slow population growth dynamics to establish a predator–prey game model with cooperative and isolated strategies. Then, we consider that the capture rate of the predator depends not only on the current but also the past instants, obtaining a new fractional-order predator–prey game model. Section 3 provides a detailed analysis for the existence, local stability, and bistability of equilibria of the model. In addition, we demonstrate that the system undergoes transcritical and Hopf bifurcations at different equilibria. In Section 4, the local stability conditions for the equilibria of the new fractional-order predator–prey game model are found, and the influence of dimension on the dynamical behavior of the system is considered. Section 5 shows numerical simulations to demonstrate the correctness of the theoretical results. Finally, the work is concluded with a discussion of results and future ideas.
2 Model Formulation
In populations, certain animals exhibit either group-oriented or solitary behaviors to acquire food resources and avoid predators, so, in this study, we consider the scenario where prey species employ distinct behavioral strategies within predator–prey systems. We categorize the prey into two subgroups: cooperative groups and solitary individuals. Prey adopting a cooperative strategy (denoted as
) act collectively to locate resources. Conversely, those employing a solitary strategy (denoted as
) forage alone. Cooperative preys can collectively defend against predators but must share resources. Solitary preys can access resources individually but face higher vulnerability to predators. At time
, the total prey population size is
, consisting of
using cooperative strategies and
using solitary strategies, such that
. The total predator population size is
.
Let
represent the frequency of the cooperative strategy among prey species, where
. Similarly,
denotes the frequency of the solitary strategy,
, and thus,
.
denotes the resource quantity available, and parameters
and
(
) signify the resource acquisition rates of cooperative and solitary prey, respectively. Values of
and
, which can be possibly different, denote the costs incurred by cooperative and solitary prey for acquiring resources and defending against predators. Based on the above framework, we can delineate the benefits associated with cooperative and solitary prey, as detailed in Table 1.
TABLE 1.
The benefits of two types prey.
Prey type
Benefits
Thus, the fitness of cooperative prey and solitary prey is as follows:
respectively, and the average fitness of the prey species is
By assessing the fitness associated with each prey strategy, we can formulate the replicator equation for the prey species as follows:
where
is the fast time scale. Suppose that the evolutionary dynamics of the prey occur rapidly compared to other processes within the predator–prey system, such as predator mortality, prey growth, predator–prey interactions, and similar factors. Given the aforementioned equations and the constraint
, we obtain
Next, we introduce the following dynamics of slow timescale population growth (for a general discussion on slow population growth, we refer to the literature [41]): (a) in the absence of predators, prey population grows logistically and (b) when predators are present, the number of prey captured increases linearly with prey biomass. Here, we consider that the capture rate of predators increases linearly with the increase of prey biomass. In the future, we can consider the Holling II (III) functional response function, which will lead to more diverse dynamic behaviors in the system. Since we consider a scenario where prey species employ distinct behavioral strategies that affect predator capture rates, we suppose that predators have capture rates of
for cooperative preys and of
for solitary preys (
). Consequently, we write
where
represents the intrinsic growth rate of the preys,
denotes the environmental carrying capacity for the preys,
indicates the conversion rate at which predators capture preys for their own growth, and
represents the natural mortality rate of the predator.
Notice that
and
. We thus have
which together with the previously derived slow and fast systems yields the following model in
:
or the predator–prey game model in
as follows:
(1)
In the above model (1), if we additionally consider that the increase in predator biomass is linked to the prey captured by those predators in the past, say at
; we can further develop our model. More specifically, the capture rate of the predator, denoted by
, is a function of the current time
and the time interval
after capturing the prey, making time-dependent extrinsic changes as well as the intrinsic changes in the capture rate of predators during their natural predation process. We then can write the following:
where
is the extrinsic capture and
is the intrinsic capture, which jointly determine the capture rate of predators. The extrinsic capture rate is influenced by environmental factors such as prey density, which determine the predator's ability to hunt at a specific time point. The intrinsic capture rate refers to the capture efficiency determined by internal factors such as physiological characteristics of predators, which is usually related to individual states and may evolve over time. Here, we use the product form to reflect the synergistic effect of internal and external factors. For example, the internal abilities of predators may be amplified or weakened by external environments, and the additive form cannot reflect this interactive influence. Suppose that there are
preys at time
. Then, during the time interval from
to
, each predator individual causes the expected increase in the predator population to be
Thus, the biomass of individuals that transition into predators at time
denoted as the flux into
is represented by
and can be recursively constructed from previous fluxes as follows:
where
represents the survival probability of a predator that appeared at time
and remains alive until time
, given a mortality rate
during this period. And
is defined by
So, we have the following:
using the properties of the exponential function. Now, let
denote a predator that transformed at time
and has not died by time 0. Then, we obtain the following:
Substituting the expression for
into
yields the following:
The predator population at time
is composed of (a) predators that were either born or arrived in the area at time
and (b) predators that were born or arrived at some earlier time
and have survived until
. Thus, the biomass of predator individuals at time
can be formulated as follows:
(2)
where
is the population of predators that were born or arrived at time
and have survived until
and thus is given by
Differentiating Equation (2) with respect to
, we have the following:
By applying the Laplace transform to Equation (4), we obtain the following:
Let
. Performing the Laplace transform on the first term of Equation (3) yields the following:
Similarly, applying the Laplace transform to the second term of Equation (3) yields the following:
where
. We assume that the intrinsic capture rate of predators is equal when faced with different strategies of prey, that is,
, then
(5)
Putting all above together, we have established the main equation describing the dynamics of the following predator population:
Using the definitions of
and the fact that
is a constant, the last equation can be written as follows:
In a similar manner, we can derive a model for the prey population:
To ensure continuity, we write
, where
is a constant and
is a Dirac delta function and arrive at the following important intermediate result:
(6)
Additionally, we hypothesize that at time
, the newly arriving predators exhibit the highest ability to capture preys, with this ability diminishing over time. For the intrinsic capture rate, we only consider the impact of predator age changes on their capture rate, and as age increases, the predator's capture rate decreases. Consequently, we propose that the predation rate follows a power law tail distribution function, in the form
When
, we know that
and
. Then system (6) reduces to system (1) and becomes a system of ordinary differential equations. In this sense, model (6) is a generalized version of model (1). Note also that for
, the Laplace transform of
of Equation (5) gives the following:
(7)
For the convenience of analyzing the integral parts in model (6), we use a generalization of the classical derivative to noninteger orders in the sense of Riemann–Liouville [42] that gives the Riemann–Liouville fractional derivative as follows:
Therefore, the fractional derivative can be written in the following convolutional form:
(8)
Using (7) and (8), we can generalize the previous result to
(9)
Substituting Equation (9) into model (6), we then obtain the model with fractional derivatives as follows:
(10)
Note that model (10) is a nonautonomous dynamical system. So, for simplicity of analysis, from now on, we assume that all parameters are constant and denote
, and
. Thus, we obtain
and finally arrive at
(11)
In what follows, we will concentrate on our newly developed models (1) and (11), respectively.
3 Dynamics of System (1)
In this section, we investigate the existence of equilibria for system (1), along with their local stability, potential bifurcations, and bistability. To facilitate our discussion in subsequent sections, we define the following quantities:
3.1 The Existence and Stability of Equilibria for System (1)
System (1) admits up to eight equilibria, which include seven boundary equilibria, respectively, denoted by
and the only interior equilibrium
.
Then, a straightforward calculation gives the following theorem.
Theorem 3.1.Regarding the equilibrium point of model (1), we have the following conclusions.
A.
The model always admits four boundary equilibria,, and
.
B.
The equilibriumexists if
.
C.
The equilibriumexists when
.
D.
The equilibriumexists if one of the following conditions holds:
a.
and;
b.
and
.
1.
The interior equilibriumexists if one of the following four conditions, andholds:
Denote the Jacobian matrix of system (1) at the equilibrium by
(12)
where the elements are as follows:
Then linear stability analysis gives the following theorems 3.2 and 3.3.
Theorem 3.2.Regarding the local stability of the equilibria identified in (A) of Theorem3.1, we have the following:
which has eigenvalues
, and
so
is an unstable equilibrium point.
Similarly, at
, the Jacobian matrix
has eigenvalues
, and
. Therefore,
is an unstable equilibrium point.
(II) The Jacobian matrix corresponding to
is
which has eigenvalues
, and
. Then, when
and
, all three eigenvalues are negative so
is a stable equilibrium point; otherwise, at least one of the eigenvalues is positive and the equilibrium is unstable.
Similarly, we can obtain (III): When
and
, the equilibrium
is stable.
Theorem 3.3.The local stability conditions of the equilibria identified in (B)–(D) of Theorem3.1are summarized as follows:
The three roots of the characteristic Equation (13) are
, and
. Since
and
, we obtain
and
. Thus, the stability of
is determined by the sign of
, implying that the equilibrium is stable if
; that is,
.
experiences a Hopf bifurcation atwhen, whereis implicitly defined by
.
Proof.Denote
(I) Let
and
be eigenvectors, corresponding to the zero eigenvalues of the Jacobian matrices
and
. Through some calculation, we have the following:
According to the expressions of
, and
, at
, we have the following:
giving
where
We have the following:
Therefore, we can write
Using the Sotomayor theorem from Reference [30] or [45 , p. 338], we find that system (1) undergoes a transcritical bifurcation at
, when
.
(II) Let
and
be eigenvectors, which correspond to the zero eigenvalues of the matrices
and
. Through some calculation, we have the following:
According to
, and
, we have the following:
Therefore,
Again, using Reference [45, Theorem 1, p. 338], system (1) undergoes a transcritical bifurcation at
when
.
(III) Let
and
be eigenvectors, which again correspond to the zero eigenvalues of the matrices
and
. Again, we have the following:
According to
, and
, we have the following:
Therefore, we arrive at
Again, Theorem 1 from [45 , p. 338] implies that system (1) undergoes a transcritical bifurcation at
when
.
(IV) In this case, we obtain the characteristic equation as
Since
is implicitly defined by
, when
, the roots of the characteristic equation show negative solution
and a pair of pure imaginary ones,
and
. Furthermore, we have the following:
so by applying the Hopf theorem ([45, Theorem 2, p. 353]), the system undergoes a Hopf bifurcation at
.
4 Dynamics of System (11)
Similar to the previous section, we now discuss the equilibria of system (11) and explore the impact of
on each equilibrium state. Next, we investigate local stability.
4.1 The Influence of
on the Equilibria of System (11)
Define the equilibrium state
by taking
Then, for model (11), taking the limit as
for both sides of (11), we obtain
(16)
We observe that there are unevaluated limits in the second and third equations of (16). In order to evaluate these limits, we perform a Laplace transform to obtain
(17)
We rewrite the above expression using a Taylor series expansion up to the second order in
and the take the inverse Laplace transform for Equation (17) to obtain
It is evident that within the long-term limit, the inverse Laplace transform of the high-order term of the Taylor expansion is zero:
Since
, we also obtain
Substituting the above analysis into expression (16), we get
(18)
For notational simplicity, we now define the following quantities:
Solving the above equations, we obtain the following eight equilibria, which include seven boundary points as follows:
The only interior equilibrium state is given by
. Let us now examine under which conditions these equilibria physically exist, as summarized in the following theorem.
Theorem 4.1.
a.
The boundary equilibriaalways exist.
b.
When, thenexists.
c.
When, thenexists.
d.
Whenandorand, thenexists.
e.
The interior equilibriumexists if one of the following conditions holds:
Among these equilibrium states,
, and
are not affected by
, but the equilibrium states
, and
are
-dependent. More precisely, we have the following:
(I) For
, we obtain
Since
, we have
, implying that
, showing that
is a decreasing function in
. From the last equation and Theorem 4.1, we also obtain the following:
i.
When
is decreasing as we increase
.
ii.
When
is not affected by
.
iii.
When
is an increasing function in
.
In Figure 1a, we plot the changes in the profile of values of equilibrium
for the prey
and the predator
on
. It is evident that the biomass of the prey decreases as
increases, whereas the biomass of the predator initially increases and then decreases with increasing
. All this shows that, when the prey adopts an isolation strategy, reducing
is beneficial for increasing prey biomass.
The evolution of the frequency of prey adopting cooperative strategy
, the biomass of prey
, the biomass of predator
as
. The selection of each parameter is as follows:
. (a)
, (b)
, (c)
. [Colour figure can be viewed at wileyonlinelibrary.com]
(II) At
, we find
Similarly,
decreases as increasing
, and
i.
If
, then
, which means that
decreases with the increase of
.
ii.
If
, then
, which means that
is not affected by
.
iii.
If
, then
, which means that
increases with the increase of
.
Similarly in Figure 1b, we show the changes in the nonisolation interior equilibrium state for the prey
and the predator
on
. Clearly, we observe that the biomass of the prey decreases with the increase of
, while the biomass of the predator first increases and then decreases with the increase of
. Therefore, in the case where the prey adopts a cooperative strategy, the reduction of
is beneficial for the increase of prey biomass.
(III) Finally, at
, we have the following:
According to the existence condition of
, when
or
is satisfied, we know that
and
, which means that
increases and
decreases with the increase of
. Furthermore, we have the following:
(a)
If
, then
, which means that
decreases with the increase of
.
(b)
If
, then
, which means that
is not affected by
.
(c)
If
, then
, which means that
increases with the increase of
.
In Figure 1c, we now plot the change in the interior equilibrium state for the cooperation frequency
, the prey
and the predator
as
. It is easy to observe that the frequency of the prey adopting cooperative strategy increases, the biomass of the prey decreases with the increase of
, while the biomass of the predator first increases and then decreases with the increase of
. Under certain conditions, an increase of
is not beneficial for the survival of preys.
When
or
is satisfied, we know that
and
, which means that
decreases and
decreases with the increase of
, and we have the following:
(a)
If
, then
, meaning that
decreases with the increase of
.
(b)
If
, then
, meaning that
is not affected by
.
(c)
If
, then
, which means that
increases with the increase of
.
4.2 Stability Analysis of System (11)
Linear stability analysis shows the following local stability of the previously found equilibrium states.
Theorem 4.2.
a.
andare always unstable.
b.
Ifand, thenis a stable equilibrium point.
c.
Ifand, thenis a stable equilibrium point.
Theorem 4.3.
a.
is a stable equilibrium point if
.
b.
is a stable equilibrium point if
.
c.
Ifandis a stable equilibrium point.
Theorem 4.4.is a stable equilibrium point if, and
.
The proofs for Theorems 4.2–4.4 are similar to those for Theorems 3.2–3.4 so we omit the details here.
Finally, we summarize the existence and local stability conditions of the equilibrium states in Table 3.
TABLE 3.
The conditions for the existence and local stability of equilibrium states.
Equilibria
Existence condition
Stability condition
Always
—
Always
and
Always
—
Always
and
and
or
and
and
or
or
or
and
5 Numerical Simulations of Typical Solutions and Behaviors
In the previous sections, we have analytically studied the local stability of the equilibrium states for the models of ordinary differential equations and fractional-order differential equations. Next, we conduct some simulations to numerically explore the effects of some key parameters. The models give rise to interesting dynamics.
5.1 Numerical Stability Analysis of System (1)
First, we demonstrate the findings identified in Theorems 3.2–3.4 through a series of figures.
Let the red, green, and blue lines represent the frequency of cooperative strategy adopted by prey over time, the biomass of prey over time, and the biomass of predators over time, respectively. In Figure 2a,b, we set the parameters as follows:
(19)
and the initial values are
and
. We find that different initial values lead the system to converge to different equilibria over time. Under other unchanged conditions, an increase in the biomass of the initial predator is beneficial for the prey to adopt cooperative strategy: At the initial time, when the prey encounters a majority of predators, the prey is more willing to adopt a cooperative strategy and engage in group activities.
Time series diagrams of the local stability of each equilibrium, showing different approaches to stability. The parameter values used for each figure are given in the text. The initial values are (a) (0.1, 7, 0.5), (b,c) (0.1, 7, 3), and (d–f) (0.1, 3, 7). [Colour figure can be viewed at wileyonlinelibrary.com]
In Figure 2c,
, and the other parameter values are consistent with (19). We see that when resources decrease, even if the resources absorption rate of cooperative strategy increases, the biomass of prey used by cooperative strategy decreases. This shows that abundant resources can promote individual preys to adopt cooperative behavior. Figure 2d uses parameters
and
, and all other parameter values are the same as specified in expression (19). In Figure 2e,
and
, and the other parameter values are consistent with (19). They all have the same initial value
. It is evident that as the conversion rate of predators increases, prey are more inclined to adopt a cooperative strategy, leading to an increase in the steady-state biomass of predators, consistent with the conclusions drawn from Figure 2a,b. In Figure 2f,
and
, and the other parameter values are consistent with (19), with identical initial value
. As the conversion rate of predators increases, the number of prey using cooperative strategies increases and the biomass of predators continues to increase.
Figure 3 shows the different bistability scenarios of system (1) in the
-space. The coordinate axes
represent the frequency of cooperative strategy adopted by the prey, the biomass of the prey, and the biomass of the predator. The blue and green lines starting from different initial values eventually converge to different equilibria: Figure 3a uses the same parameter values as in (19), while the parameter values in Figure 3b,c are
and
, respectively. All the other parameter values are consistent with (19).
The phase diagram of equilibrium point bistability in
-space. The blue and green lines starting from different initial values ultimately converge to different equilibria to demonstrate various dynamics of bistability. The coordinate axes represent the frequency of cooperative strategy adopted by the prey, the biomass of the prey, and the biomass of the predator. [Colour figure can be viewed at wileyonlinelibrary.com]
In Figure 3a, under the same parameter values, different initial conditions eventually tend toward
and
: When the initial frequency of prey adopting cooperation and the initial biomass of predators are small, over time, the prey adopting cooperative strategy in the population eventually disappears, and predators do not exist. Only preys that adopt an isolation strategy can survive. When the initial biomass of the predator is high or the initial frequency of cooperative strategy adopted by the prey is high, the preys in the population ultimately choose a cooperative strategy to resist the capture of the predator: Predators disappear, only cooperative preys survive. In Figure 3b, different initial values eventually tend toward
and
: Under certain initial conditions, predators and prey coexist. In Figure 3c, different initial values ultimately stabilize at
and
. When the absorption rate of resources by cooperative and isolated preys is changed, preys and predators coexist in the population, and all preys adopt the same strategy, which is influenced by the initial values of the three variables. Overall, Figure 3 confirms the results of Theorem 3.5. This also indicates that different initial and parameter values can elicit a rich bistability scenario to the system. Moreover, we find that an increase in the conversion rate of predators is beneficial for their survival.
Using Figure 4, we demonstrate the effects of changes in cooperation cost
and isolation cost
on the frequency
of cooperation adopted by prey, the biomass of prey
, and the biomass of predator
. When the cooperation cost
is small, the change of isolation cost cannot affect the final stability of the system. In this situation, it is easy to observe that only the prey population survives in the population, and the prey tends to adopt a cooperative strategy, while the predator becomes extinct. When the cooperation cost
increases, the increase of the isolation cost
promotes the adoption of cooperative strategy by the prey, and the biomass of the prey increases while the biomass of the predator decreases. If the isolation cost is fixed, the increase of cooperative cost inhibits the adoption of cooperative behavior by the prey, and the biomass of the prey decreases while the biomass of the predator increases.
The joint impact diagram of cooperative prey cost
and isolated prey cost
in system (1). Panel a shows the change of the frequency
of prey adopting cooperative strategy as a function of
and
, panel b shows the change of the biomass
of the prey as a function of
and
, and panel c shows the change of the biomass
of the predator as a function of
and
. The other parameter values in the figure are consistent with Figure 2f. [Colour figure can be viewed at wileyonlinelibrary.com]
Figure 5 reveals the effects on the changes of the absorption rate
of cooperative prey and the absorption rate
of isolated prey for the frequency
of cooperation adopted by prey, the biomass of prey
, and the biomass of predator
. Due to practical considerations, we only observe the upper left part of each figure. When the absorption rate of resources by cooperative preys is high, predators disappear from the population, and only preys that adopt cooperative strategy exist. Instead, the absorption rate of cooperative prey on resources remains unchanged, as the absorption rate of isolated prey on resources increases and the biomass of predators in the population increases. If the absorption rate of isolated preys on resources remains unchanged, as the absorption rate of cooperative preys on resources increases, the biomass of predators in the population decreases and disappears, while the biomass of prey using a cooperative strategy increases. This allows to conclude that increasing the absorption rate of isolated prey on resources is beneficial for the survival of predators.
The joint impact diagram of the absorption rate
of cooperative prey on resources and the absorption rate
of isolated prey on resources in system (1). Panel a shows the change of the frequency
of prey adopting cooperative strategy as a function of
and
, panel b shows the change of the biomass
of the prey as a function of
and
, and panel c shows the change of the biomass
of the predator as a function of
and
. The other parameter values in the figure are consistent with Figure 2f. [Colour figure can be viewed at wileyonlinelibrary.com]
Figure 6 shows the impact of changes in predator conversion rate
on the frequency
of cooperative strategy adopted by preys, as well as the biomass of preys
and predators
. Each subfigure is divided into three parts: As the conversion rate
increases, the stability of the equilibria changes, where we only mark the corresponding stable equilibrium points. Specifically, when
, only
is stable, while
and
are unstable. Conversely, when
, only
is stable, while
and
are unstable. The equilibrium points
and
interchange their stability at
, resulting in a transcritical bifurcation of the system. Similarly, when
, only
is stable, while
and
are unstable; the equilibrium points
and
also interchange their stability at
, indicating that the system undergoes a transcritical bifurcation. This implies that as the conversion rate of predators increases, the frequency of prey adopting cooperative strategies escalates, the biomass of predators increases, and the biomass of prey declines. It is important to emphasize that a high conversion rate of predators benefits their survival and encourages prey to adopt cooperative strategies to resist predation.
Schematic diagram of changes in
, and
with predator conversion rate
. Each subfigure is divided into three parts, and the stability of the equilibrium point of the system changes with the increase of
. The other parameter values in the figure are consistent with Figure 2f. [Colour figure can be viewed at wileyonlinelibrary.com]
Figure 7 shows the impact if resources
instead change on the frequency
of the cooperative strategy adopted by prey, as well as the biomass of prey
and predator
. First, when
, as resources increase, the biomass of prey and prey using cooperative strategy increases, while the biomass of predators decreases. Then, when
, the system undergoes Hopf bifurcation, and the interior equilibrium point becomes unstable. As resources increase,
is stable, predators become extinct and only preys that adopt cooperative strategy exist in the population. This let us conclude that abundant resources are beneficial for the survival of prey, and prey population is more willing to adopt a cooperative strategy. On the contrary, abundant resources are not conducive to the survival of predators, and an increase of resources can lead to the extinction of predators.
Bifurcation diagrams showing the changes in
and
with resources
. Each subfigure is divided into three parts, and the system experiences a Hopf bifurcation when
. The other parameter values in the figure are consistent with Figure 2f. [Colour figure can be viewed at wileyonlinelibrary.com]
The results and phenomena discussed in this section are contingent upon the parameter values selected for the analysis. Consequently, the following conclusions should be interpreted with the understanding that they may not be universally applicable to all possible parameter configurations: Under the parameter values illustrated in Figure 2c,f, both cooperative and isolated strategies can coexist within prey species over time. Moreover, both prey and predator populations can coexist, sustaining their biomasses at stable levels. In Figure 6, an increase in the conversion rate of predators leads to a greater willingness among prey to adopt cooperative strategies to resist predation, resulting in an increase in predator biomass during this period. Additionally, in Figure 7, abundant resources not only encourage the adoption of cooperative strategies within prey species but can also lead to a decline in predator biomass and an upsurge in prey biomass. Furthermore, the initial values of prey cooperation frequency, prey biomass, and predator biomass significantly influence the final convergence state of the system, as depicted in Figure 3. Under specific conditions, the system may exhibit richer and more complex dynamics, such as bistability, transcritical bifurcation, and oscillations resulting from a Hopf bifurcation. Variations in system parameters can affect these dynamics in diverse ways.
5.2 The Impact of
on System (11)
Now we consider the impact of
on the equilibrium state of system (11), with different parameter values.
In Figure 8, we present the time series of
, and
as
changes, and the initial value is
. The fractional-order model (11) becomes the ODE model (1) when
. Obviously, the frequency
of prey using cooperative strategy increases sharply in a short period of time and then slows down to a steady state. The effect of
depends on the initial state of the system. The steady-state value of
increases with the increase of
. When
, the biomass of the prey rapidly increases to a steady state in a short period of time, while when
, the biomass
of the prey increases sharply and then decreases to a steady state. As
increases, the steady-state value of
decreases. When
, the biomass of the predator sharply decreases to a steady state in a short period of time, while when
, the biomass
of the predator sharply decreases and then increases to a steady state. As
increases, the steady-state value of
increases. Parameter
affects the capture rate of predators; that is, as
increases, the capture rate of predators in the population also increases. So we can find that as the capture rate of predators increases, the frequency of cooperative strategies adopted by preys in the population increases. The preys are more willing to adopt a cooperative strategy to resist the capture by predators, and the biomass of the prey in the population decreases while the biomass of predators increases. Based on the above description, in this case, we conclude that the decrease of the fractional-order
is beneficial for the reduction of predator biomass.
The time series diagram of
and
varies with
(
and 1). The other parameter values are
. [Colour figure can be viewed at wileyonlinelibrary.com]
In Figure 9, we present the time series figures of
, and
as
changes. The initial value is
. Under different conditions, the phenomena in Figure 9 are similar to that in Figure 8, and we conclude that the decrease of the fractional-order
leads to the extinction of predators.
The time series diagram of
and
varies with
(
and 1). The other parameter values are
. [Colour figure can be viewed at wileyonlinelibrary.com]
In Figure 10a, when
, the frequency
of the prey adopting cooperative strategy rapidly decreases to a steady state
, while when
, the frequency
increases to a steady state
. This indicates that fractional-order
promotes the preys not adopting cooperative strategy. In Figure 10b,c, when
, the biomass
of the prey and the biomass
of the predator rapidly increase to a steady-state
and rapidly decrease to 0, respectively. When
, the biomass of the prey first decreases and then slowly increases to a steady-state
, while the biomass of the predator slowly decreases to steady-state 0. Fractional-order
promotes a rapid increase in the prey species and a rapid decrease in the predator species: It is evident that fractional-order
can encourage prey to adopt an isolation strategy in this case.
The time series diagram of
, and
varies with
(
, and 1). The other parameter values are
. [Colour figure can be viewed at wileyonlinelibrary.com]
Finally, in Figure 11, we can see that, as
changes, the three variables eventually tend to the same steady state, but the convergence speed is different. In Figure 11a, as
decreases, the frequency of adopting cooperative strategies for prey reaches a steady state at a slower rate. In Figure 11b, as
decreases, the rate at which the biomass of the prey reaches a steady state shows faster dynamics. In Figure 11c, as
decreases, the rate at which the biomass of the predator reaches a steady state is also faster. Therefore, in this case, we conclude that
does not affect the steady state to which the system eventually converges to, but it does affect the speed of convergence.
The time series diagram of
, and
varies with
(
, and 1). The other parameter values are
. [Colour figure can be viewed at wileyonlinelibrary.com]
6 Conclusions and Discussions
In this article, we consider two optional strategies among prey species: a cooperative strategy and an isolation strategy. The prey that adopts a cooperative strategy must share the acquired resources and can jointly defend against predators, while the prey that adopts an isolation strategy can enjoy the resources alone and face the capture of predators alone. One of the purposes of this article is to explore the evolutionary impact of prey strategies on the dynamics of predator–prey systems, using fast time game dynamics and slow time population growth dynamics. The replicator equation shows the frequency variation of cooperative strategy in the prey species, and the two population growth equations describe the biomass changes of prey and predator populations.
While the results and phenomena presented in Section 5 provide valuable insights into the system's dynamics, they emerge under specific conditions, as determined by the parameter values chosen for the analysis. Therefore, these conclusions should be interpreted with the understanding that they are not universally applicable to all potential parameter configurations.
We also consider the impact of past predator capture prey on the current predator biomass in the predator–prey game system, which results in a new fractional-order predator–prey game model. We derive the new model through a potential physical stochastic process with CTRW, which is different from previous fractional-order models. We consider the local stability of each equilibrium state in the new fractional-order predator–prey game model and the impact of the fractal dimension
on system dynamics. The fractal dimension influences the capture rate of predators: As the fractal dimension increases, the capture rate of predators within the population also rises. Consequently, under the parameter values in Figures 8 and 9, as the capture rate of predators increases, the frequency of cooperative strategies adopted by the prey also rises. Preys are more inclined to engage in cooperative behavior to resist predation. This leads to a decrease in the prey biomass within the population, while the biomass of predators increases. In short, we find that
can promote a decrease or disappearance in the number of prey adopting a cooperative strategy and lead to the extinction of predators. Also, in Figure 11, we see that the fractal dimension does not affect the steady state to which the system eventually converges but does affect the speed of convergence. In other words, by changing the size of fractal dimension (the predator's capture rate changes) in the system, the frequency of cooperative strategies adopted by the prey, the biomass of the preys, and the biomass of the predators ultimately stabilize toward the same steady state, but their rates of approaching the steady state are different. This can have important biological consequences.
Different to the existing literature, this article introduces the influence of individual behavioral dynamics of prey in the dynamics of predator–prey populations and combines the replicator equation with the population growth equation. In addition, based on the aforementioned predator–prey game model, we consider the impact of past predator capture prey on the current predator biomass and derives a novel fractional-order predator–prey game model. In the real world, the capture rate of predators within a population is not constant and varies over time, likely decreasing with age. Thus, in our study here, we characterize this variation in predator capture rate over time through a CTRW process and construct a predator–prey game system with fractional-order terms, which in our opinion better reflects real-world considerations and more accurately represents the phenomenon of the varying predator capture rates. Additionally, we observe that changing the amount of deployed resources can control the behavior of preys (cooperation or isolation) and regulate the biomass of both preys and predators. Therefore, these models provide indications to establish effective mechanisms to control individual behavior and the biomass of species.
This work establishes the foundation for future related research and has important significance. However, a possible limitation is given by the capture rate of predators on prey, which increases linearly with the increase of prey biomass. Next, we could consider Holling II (III) functional response functions [46, 47] as they should lead to more diverse dynamic behaviors in the system. In addition, the limit process of a CTRW with power law waiting time can lead to fractional diffusion equations [48] and fractional reaction–diffusion equations [49-51]. Therefore, we can also explore spatial–temporal evolutionary games [19], to enrich the dynamical scenarios that are possible.
This work is supported by the National Natural Science Foundation of China (no. 12271308), the China Scholarship Council (no. 202308370307), and the SDUST Innovation Fund for Graduate Students (YC20210231). Open access publishing facilitated by Swinburne University of Technology, as part of the Wiley - Swinburne University of Technology agreement via the Council of Australian University Librarians.
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