Hopf–Zero bifurcation in an age-dependent predator–prey system with Monod–Haldane functional response comprising strong Allee effect

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Abstract

The article is intended to study the Hopf−Zero bifurcation of a novel age-dependent predator−prey system with Monod−Haldane functional response comprising strong Allee effect. Here in this system the predators fertility function f(x) is regarded as a piecewise function with respect to their maturation period τ. The system is interpreted as a non-densely defined abstract Cauchy problem, and the condition of the existence and uniqueness of the non-negative steady state for a coupled dynamic system both a partial differential equation and an ordinary differential equation is yielded. Via employing the center manifold theorem and normal form theory for semilinear equations with non-dense domain, we discover much richer fresh dynamical behavior in an age-dependent predator−prey system than the existing ones.

Introduction

In 1798, Malthus, a British demographer, put forward the world-famous Malthusian species system during his tenure as a priest. At this time, the species presents exponential growth, but the growth of species is always affected by environment, food, natural resources and other factors, and cannot grow indefinitely. Therefore, in 1938, in order to be more realistic and relevant, Verhulst, a Dutch biologist, introduced environmental capacity and proposed a Logistic system. This system shows that when the initial number of species is less than the surroundings capacity, the final size of population increases monotonously and tends to the environmental capacity. Conversely, its final biomass decreases monotonously and escapes to the carrying capacity. It can be seen that the initial scale of species determines its dynamic properties. For many species, a small amount is not conducive to the survival of the species, such as difficult spouses, inbreeding, and this appearance is known as the Allee effect. This phenomenon first appeared in 1938. Allee, a famous American ecologist, raised the question in document [1]: What is the minimum number of natural species that can survive? He also discussed the significant impact of crowding on demography and life quality, and found that the growth rate of small density species was not always positive, nor could it be as decreasing as Logistic system. Its typical form isdW(t)dt=rW(t)[1W(t)N][W(t)M], where the species of individuals at time t is described as W(t); r is the intrinsic growth rate of species; N is the circumstances capacity. To get closer to reality, we will give two scenarios below:

1. When 0<M<N, M is the minimum amount of species survival, which is so-called “threshold” value [2], [3]. That is to say, such growth has strong Allee effect;

2. While M<0<N, there is no threshold value at this time, which is called weak Allee effect [4], [5]. See Fig. 1 for all of the above.

Besides, since no species exists in isolation, but is closely related to other species, the interaction between these species is mainly manifested in three different types: Competitive system, reciprocal system and predator−prey system. We pay attention to the predator−prey system in that the competition system and reciprocity system have been perfected [6], [7]. In 1925 and 1926, American physical chemist Lotka and Italian mathematician Volterra successively applied a set of ordinary differential equations in their book and work to research the interaction between the predator and prey [8], [9]. The authors adopt that the Lotka−Volterra functional response function which is a straight line through the origin and is unbounded refers to the change in the density of the prey attached per unit time per predator as the prey density changes. Nevertheless, this is too idealistic and a little unreasonable, in the meantime, a bit inconsistent with reality. Thus, more reasonable functional response functions should be nonlinear and bounded. In 1913, German-American chemist Michaelis and his assistant Menten brought forward the following functional response function:g[W(t)]=mW(t)a+W(t), here m>0 stands for the maximal growth rate of species and a>0 is the half-saturation constant. In 1959, Holling [10] also used this functional response function as one of the predator functional response functions. It is now called a Michaelis−Menten functional response function or a Holling type II functional response function. Another kind of functional response function isg[W(t)]=mW2(t)a+bW(t)+W2(t), which is referred to a sigmoidal functional response function (i.e. a generalized Holling type III functional response function), while the simplificationg[W(t)]=mW2(t)a+W2(t) is known as a Holling type III functional response function. For other types of functional response functions, we refer to [11]. Furthermore, experiments [12], [13] have shown that there is a non-monotonic functional response function at the microbial level: When nutrient concentration reaches a higher level, it may inhibit specific growth rate. To simulate this suppression, Andrews [14] considers a functional response function as follows:g[W(t)]=mW(t)a+bW(t)+W2(t) called the Monod−Haldane functional response function (i.e. a generalized Holling type IV functional response function), while the simplificationg[W(t)]=mW(t)a+W2(t) is termed a Holling type IV functional response function.

For predator and prey species, a functional response is the intake rate of a predator as a function of predator density, and it is usually associated with the numerical response, which is the reproduction rate of a predator as a function of prey density, see Fig. 2.

In addition, partly because immature predators cannot reproduce as they eat some preys, age-dependent fertility is one of the most important parameters in the theory and application of species dynamics. And so, some age-dependent predator−prey systems have been investigated [15], [16], [17], [18], [19], [20], [21], [22], [23], [24]. One part of these systems concentrate on the existence, boundedness and stability of solutions by making use of the characteristics method [15], [16], [17] or variational way [19]. The other part of these systems focus on the non-trivial periodic solution through applying Hopf bifurcation theorem [25]. Though, with a broad and profound understanding of age structure, at the same time, motivated by the above literatures, we will deliberate the existence of the Hopf−Zero bifurcation of a new age-dependent predator−prey system with Monod−Haldane functional response comprising strong Allee effect by taking advantage of the center manifold theorem [26] and normal form theory [27], [28] for semilinear equations with non-dense domain, which have never been discussed previously in the existing references.

In this regard, set p(t,x) be the total species density of the predator at time t with age x; W(t) represent the total species density of the prey at time t. Nowadays, we consider the coupled dynamic system both a partial differential equation and an ordinary differential equationp(t,x)t+p(t,x)x=μp(t,x),dW(t)dt=rW(t)[1W(t)N][W(t)M]mW(t)0+p(t,x)dxa+bW(t)+W2(t) with boundary conditionp(t,0)=nmW(t)0+f(x)p(t,x)dxa+bW(t)+W2(t), initial conditionsp(0,x)=p0(x)L+1((0,+),R),W(0)=W0[0,+). All the parameters of system (1.1) and their biological significance are listed in Table 1. In the remainder of our paper, the predators fertility function f(x) satisfies Postulation 1.1.

Postulation 1.1

There is no loss in generality in supposing thatf(x)={f,xτ,0,x(0,τ),and0+f(x)eμxdx=1, where τ is the maturity period of the predator species and τ>0, f>0.

The structure of the work is organized as below. In the next Section, we make some preliminary results. In Section 3, we do some analyses of the system. In Section 4, we discuss the existence condition of Hopf-Zero bifurcation of the system. In the last Section, we make a summary.

Section snippets

Rescaling time and age

Let tˆ=tτ, xˆ=xτ, then pˆ(tˆ,xˆ)=τp(τtˆ,τxˆ),Wˆ(tˆ)=W(τtˆ). For the sake of convenience, we drop the hat notation. Proceed to the next step, we havep(t,x)t+p(t,x)x=τ[μp(t,x)],dW(t)dt=τ{rW(t)[1W(t)N][W(t)M]mW(t)0+p(t,x)dxa+bW(t)+W2(t)} with boundary conditionp(t,0)=τnmW(t)0+f(x)p(t,x)dxa+bW(t)+W2(t), initial conditionsp(0,x)=p0(x)L+1((0,+),R),W(0)=W0[0,+), in which the age-specific fertility sectioned function f(x) becomesf(x)=f1[1,+](x)={f,x1,0,otherwise,andf=μeμτ.

Conversion of a non-densely defined abstract Cauchy problem

Denote S(t

Existence and uniqueness of coexistence steady state

Suppose v(x)=[0R2u(x)] be a steady state of system (2.3), thenF[0R2u(x)]+τK[[0R2u(x)]]=0,[0R2u(x)]D(F), equivalently,u(0)+τQ[u(x)]=0,u(x)τCu(x)=0. On the basis of Eq. (3.1), we concludeu(x)=[p(x)s(x)]=[τnmW0+f(x)p(x)dxa+bW+W2eτμxτ[r(M+N)NW2rNW3mW0+p(x)dxa+bW+W2]eτrMx], where W=0+s(x)dx. Through solving Eq. (3.2), we derive Lemma 3.1.

Lemma 3.1

A non-densely defined abstract Cauchy problem (2.3) has always critical pointsv1(x)=[0R2[0L1((0,+),R)0L1((0,+),R)]],v2(x)

Existence of the purely imaginary root and zero root

It follows from Eq. (3.7) thatUτ(η)=η2+Θη+Φ+(Ωη+Ψ)eτη. According to Uτ(0)=Φ+Ψ=0,dUτ(η)dη|η=0=Θ+ΩτΨ0 for any τ>0, we know that η=0 is a root of Eq. (4.1) with multiplicity 1.

Put η=σi(σ>0) be a purely imaginary root of the equation Uτ(η)=0, thenUτ(σi)=σ2+Θσi+Φ+(Ωσi+Ψ)eτσi=0. Through segregating the real part and imaginary part of Eq. (4.2), we provideσ2Φ=Ωσsin(στ)+Ψcos(στ),Θσ=Ψsin(στ)Ωσcos(στ). Let me put it another way,sin(στ)=Ωσ(σ2Φ)+ΨΘσΨ2+Ω2σ2,cos(στ)=Ψ(σ2Φ)ΘΩσ2Ψ2+Ω2σ2.

Summary

In this article, we study an age-dependent predator−prey system where the effects of the Monod−Haldane functional response and strong Allee effect are considered. Nonetheless, we do not pay attention to the effects of other non-monotonic functional response functions and weak Allee effect on the age-dependent predator-prey systems. Moreover, we find that this system occurs the Hopf-Zero bifurcation near the coexistence steady state while the bifurcation parameters λ=(λ1,λ2)=(0,0), which is the

Acknowledgements

We would like to express our gratitude to the anonymous editors and reviewers for carefully reading the manuscript and for valuable comments and suggestions which considerably improved the presentation of the article. This work was partially supported by NSF of P. R. China (11571382).

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