Open Access

Stability and Hopf bifurcation of a producer-scrounger model with age-structure

Advances in Difference Equations20162016:243

https://doi.org/10.1186/s13662-016-0968-2

Received: 31 March 2016

Accepted: 8 September 2016

Published: 22 September 2016

Abstract

We derive a producer-scrounger model with age-structure in scrounger and looks into its dynamics. Using the methods of eigenvalue analysis and Lyapunov function, we find sufficient and necessary conditions for globally asymptotical stability of extinction equilibrium and scrounger-free equilibrium. A so-called basic reproduction ratio \(R_{0}\) was established to determine whether the scrounger is extinct or uniformly persistent. It is found that if \(R_{0}>1\), the mature time τ does change the dynamical behavior of the model. We confirm that Hopf bifurcation happens if the mature time τ increases.

Keywords

producer-scrounger model age-structure stability uniform persistence Hopf bifurcation

1 Introduction

There are three forms of basic interaction between two species: cooperation, competition, and predator-prey, which has been extensively investigated (e.g., see [1, 2] and the references therein). However, interaction forms between species in nature are complex and diverse. To forage foods, some species not only catch and feed on prey, but also can scrounge foods from others. For example, Bugnyar and Kotrschal [3] found that the free-ranging ravens steal wolves’ food in a game park and the wolves prevent their food from being stolen. This phenomenon is called kleptoparasitism and describes an interaction form between two species, scrounger and producer. Here, scrounger steals food from producer.

To describe the kleptoparasitism interaction, Cosner and Nevai [4] recently established the following diffusive model:
$$ \textstyle\begin{cases} \frac{\partial P}{\partial t}=d_{1}\Delta P+(-b_{1}-a_{1}P)P+m(x)\frac {dP}{S+d}, \quad t\ge0, \\ \frac{\partial S}{\partial t}=d_{2}\Delta S+(-b_{2}-a_{2}S)S+\theta m(x)\frac{PS}{S+d},\quad t\ge0. \end{cases} $$
(1.1)
Here \(x\in\Omega\), Ω is a bounded domain in \(\mathbb{R}^{n}\) with smooth boundary, P and S are the densities of producer and scrounger, respectively, \(m(x)\) denotes the producer’s ability to discover food at \(x\in\Omega\), and \(d>0\) represents the producer’s ability to avoid food to be stolen, \(b_{i}\) is the natural mortality rate, \(a_{i}\) is the death rate caused by intraspecies competition, \(d_{i}>0\) is the spatial diffusive rate, and θ is the energy transition rate. Note that letting \(S=0\) in the first equation of (1.1), we have
$$\frac{\partial P}{\partial t}=d_{1}\Delta P+\bigl[m(x)-b_{1}-a_{1}P \bigr]P, $$
and thus we can regard \(m(x)\) as the birth rate of producer without scourging. Furthermore, \(m(x)\frac{d}{S+d}\) is the birth rate of producer with scounging.

General speaking, all individuals always have two stages, immature and mature, and they perform very differently in some aspects. For example, the immature ravens are incapable to fly and have to stay in the nest while the mature ravens undertake the task of searching for food. In mathematics, a model with delay τ (being the maturation period) can be established to describe the population dynamics with age-structure (e.g., see [513]). In the current paper, we try to establish a producer-scrounger model with age-structure in scrounger, and study the dynamics and the influence of maturation period in the scrounge interaction.

The paper is organized as follows. In Section 2, a producer-scrounger model with age-structure in scrounger is derived. In Section 3, the basic theory, including existence, uniqueness, positivity and boundedness of solutions for the model are discussed. The stability of equilibriums, uniform persistence, and Hopf bifurcation are investigated in Section 4. Some numerical simulations and concluding discussions are given in the last section.

2 Model derivation

Let \(P(t)\) remain the density of producer at time \(t\ge0\), and positive numbers \(b_{1}\), \(a_{1}\), m, d, and θ be of the same meaning as in system (1.1). Assume that \(S(t)\) and \(I(t)\) express the densities of mature scrounger and immature scrounger, respectively, \(b_{2}\) (\(b_{3}\)) is the natural mortality rate of mature (immature) scrounger, \(a_{2}\) (\(a_{3}\)) is the death rate of mature (immature) scrounger caused by intraspecies competition. Furthermore, assume that there exists neither competition between the immature scrounger and mature scrounger nor intraspecies competition between the immature scroungers (i.e., \(a_{3}=0\)). For simplicity, we assume that the spatial environment is homogeneous, and thus \(m(x)\equiv m\) is a constant.

Firstly, we see that the producer fits an equation as follows:
$$\frac{dP(t)}{dt}=-b_{1}P(t)-a_{1}P^{2}(t)+m \frac{dP(t)}{S(t)+d}. $$
Now we derive equations for scrounger. Let \(i(t,a)\) be the density of immature scrounger at time t and age a. Then \(i(t,a)\) satisfies
$$ \frac{\partial i(t,a)}{\partial t}+\frac{\partial i(t,a)}{ \partial a}=-b_{3}i(t,a). $$
(2.1)
Let \(I(t)=\int_{0}^{\tau}i(t,a)\, da\), where \(\tau\geq0\) is a constant number that denotes the time period for an immature to become mature from its birth. Integrating equation (2.1) from 0 to τ with respect to the variable a, we get
$$ \frac{dI(t)}{dt}=i(t,0)-i(t,\tau)-b_{3}I(t). $$
(2.2)
Here, \(i(t,0)\) denotes the density of new-born scrounger at time t, and \(i(t,\tau)\) is the recruitment term, coinciding with those of maturation age.

Suppose that there is a resource of one. If the proportion of this resource ultimately acquired by the producers is \(\frac{d}{S(t)+d}\), and the mature scroungers have resource accounted for \(1-\frac {d}{S(t)+d}=\frac{S(t)}{S(t)+d}\). Considering the producer’s ability to discover food, the number of producers, and the energy transition rate, we can regard \(\theta m\frac{P(t)}{S(t)+d}\) as the birth rate of mature scrounger, and then \(i(t,0)=\theta m\frac{P(t)}{S(t)+d}S(t)\).

To calculate \(i(t,\tau)\), denote \(i^{\xi}(a)=i(a+\xi,a)\). Then
$$ \frac{di^{\xi}(a)}{da}=-b_{3}i^{\xi}(a). $$
(2.3)
Solving (2.3), we get
$$ i^{\xi}(a)=i^{\xi}(0)e^{-b_{3}a}. $$
(2.4)
Let \(a=\tau\) and \(\xi=t-\tau\) in (2.4). Then
$$ i(t,\tau)= i(t-\tau,0)e^{-b_{3}\tau} =\theta m\frac{P(t-\tau)S(t-\tau)}{S(t-\tau)+d}e^{-b_{3}\tau}. $$
Therefore, equation (2.2) can be rewritten as
$$ \frac{dI(t)}{dt}= -b_{3}I(t)+\theta m\frac{P(t)S(t)}{S(t)+d} -\theta m \frac{P(t-\tau)S(t-\tau)}{S(t-\tau)+d}e^{-b_{3}\tau}, $$
and further the equation about the mature scrounger follows:
$$ \frac{dS(t)}{dt}= -b_{2}S(t)-a_{2}S^{2}(t)+ \theta m\frac{P(t-\tau )S(t-\tau)}{S(t-\tau)+d}e^{-b_{3}\tau}. $$
Summarizing the discussion, we obtain the following producer-scrounger model with age-structure:
$$ \textstyle\begin{cases} \frac{dP(t)}{dt}=-b_{1}P(t)-a_{1}P^{2}(t)+m\frac{dP(t)}{S(t)+d}, \\ \frac{dI(t)}{dt}=-b_{3}I(t)+\theta m\frac{P(t)S(t)}{S(t)+d}-\theta m\frac{P(t-\tau)S(t-\tau)}{S(t-\tau)+d}e^{-b_{3}\tau}, \\ \frac{dS(t)}{dt}=-b_{2}S(t)-a_{2}S^{2}(t)+\theta m\frac{P(t-\tau )S(t-\tau)}{S(t-\tau)+d}e^{-b_{3}\tau}. \end{cases} $$
(2.5)
Note that in (2.5), \(I(t)\) is independent of the first and third equations. Therefore, we only need to study the following system with two equations:
$$ \textstyle\begin{cases} \frac{dP(t)}{dt}=-b_{1}P(t)-a_{1}P^{2}(t)+m\frac{dP(t)}{S(t)+d}, \\ \frac{dS(t)}{dt}=-b_{2}S(t)-a_{2}S^{2}(t)+\theta m\frac{P(t-\tau )S(t-\tau)}{S(t-\tau)+d}e^{-b_{3}\tau}. \end{cases} $$
(2.6)

Remark 2.1

If \(\tau=0\), then system (2.5) yields to (1.1) with homogeneous spatial environment.

Remark 2.2

If there exists competition between immature scroungers (i.e., \(a_{3}>0\)), then (2.1) is replaced by
$$\frac{\partial i(t,a)}{\partial t}+\frac{\partial i(t,a)}{\partial a}=-b_{3}i(t,a)-a_{3}i(t,a)I(t). $$
Similar analysis leads to a model
$$ \textstyle\begin{cases} \frac{dP(t)}{dt}=-b_{1}P(t)-a_{1}P^{2}(t)+m\frac{dP(t)}{S(t)+d}, \\ \frac{dI(t)}{dt}=-b_{3}I(t)-a_{3}I^{2}(t)+\theta m\frac{P(t)S(t)}{S(t)+d} \\ \hphantom{\frac{dI(t)}{dt}={}}{}-\theta m\frac{P(t-\tau)S(t-\tau)}{S(t-\tau)+d}\exp [-\int _{t-\tau}^{t} (b_{3}+a_{3}I(\eta) )\,d\eta ], \\ \frac{dS(t)}{dt}=-b_{2}S(t)-a_{2}S^{2}(t) +\theta m\frac{P(t-\tau)(t-\tau)}{S(t-\tau)+d}\exp [-\int _{t-\tau}^{t} (b_{3}+a_{3}I(\eta) )\,d\eta ]. \end{cases} $$
(2.7)

3 Basic theory of solutions

In this section, we study the existence, uniqueness, positivity, and boundedness of solutions of (2.6).

Denote \(X=C([-\tau,0],\mathbb{R}_{+}^{2})\), \(\mathbb{R}_{+}^{2}=\{ (x,y)^{T}:x\ge0,y\ge0\}\). Then X is a Banach space with norm \(\|\phi \|=\max\{|\phi(\xi)|:\xi\in[-\tau,0]\}\), where \(|\cdot|\) is the standard norm in \(\mathbb{R}^{2}\). For \(\sigma>0\) and \(\phi\in C([-\tau,\sigma),\mathbb{R}^{2})\), define \(\phi_{t}\in X\) as \(\phi _{t}(\xi)=\phi(t+\xi)\) for \(t\in[0,\sigma)\) and \(\xi\in[-\tau,0]\).

Define \(f: X\rightarrow\mathbb{R}^{2}\) by
$$f(\phi)=\left ( \textstyle\begin{array}{@{}c@{}} -b_{1}\phi_{1}(0)-a_{1}\phi_{1}^{2}(0)+m\frac{d\phi _{1}(0)}{\phi_{2}(0)+d} \\ -b_{2}\phi_{2}(0)-a_{2}\phi_{2}^{2}(0)+\theta m\frac{\phi_{1}(-\tau)\phi _{2}(-\tau)}{\phi_{2}(-\tau)+d}e^{-b_{3}\tau} \end{array}\displaystyle \right ), $$
where \(\phi=(\phi_{1},\phi_{2})^{T}\in X\). It is easy to see that f is continuous and Lipschitz on ϕ in every compact set of X. In view of the basic theory of functional differential equations ([14], Theorem 2.2.3), we know that for any initial function \(\phi =(\phi_{1},\phi_{2})^{T}\in X\), there exists a unique local solution \((P(t),S(t))^{T}=(P(t;\phi),S(t;\phi))^{T}\) of (2.6).
For biological significance, we consider the initial function satisfying
$$ \begin{aligned} &P(\xi)=\phi_{1}(\xi)\ge0, \qquad S(\xi)=\phi_{2}(\xi)\ge0, \quad \xi\in [-\tau,0], \\ &\phi_{1}(0)>0,\qquad \phi_{2}(0)>0. \end{aligned} $$
(3.1)
In the discussion of boundedness of solutions, we need the positivity of the immature scrounger. Therefore, in the following, to express the initial value condition of immature scrounger, we use
$$I(\xi) = \phi_{3}(\xi)\geq0,\quad \xi\in[-\tau, 0],\qquad \phi_{3}(0) > 0. $$
Then we have the following conclusion.

Lemma 3.1

(Positivity)

The solution \((P(t),S(t),I(t))^{T}\) of (2.5) with initial function \(\phi\geq\mathbf{0}\) is nonnegative on its existence interval. Furthermore, if \(\phi_{1}(0)>0\), \(\phi_{2}(0)>0\), then \((P(t),S(t))> (0,0)\) as long as the solution \((P(t),S(t))\) exists.

Proof

Assume that the maximal existing interval of \((P(t),S(t),I(t))\) is \([0,T)\). Without loss of generality, we may assume that \(\tau< T\). For \(t\in[0,\tau]\), we have \(P(t-\tau)\ge0\) and \(S(t-\tau)\ge0\). Thus, for \(t\in[0,\tau]\), we have
$$\frac{dS(t)}{dt}=-b_{2}S(t)-a_{2}S^{2}(t)+\theta m \frac{P(t-\tau)S(t-\tau )}{S(t-\tau)+d}e^{-b_{3}\tau}\ge-b_{2}S(t)-a_{2}S^{2}(t). $$
Comparing \(S(t)\) with the solution \(V(t)\) of the initial problem
$$\frac{dV(t)}{dt}\geq-b_{2}V(t)-a_{2}V^{2}(t), \qquad V(0)=S(0)\geq0, $$
we have \(S(t)\geq0\) for \(t\in[0,\tau]\). By using the induction argument, \(S(t)\geq 0\) holds as long as it exists.
Assume that there exists \(t_{0}\) such that \(P(t_{0})<0\). Since \(P(0)\geq 0\), by continuity and without loss of generality, we can find some \(t_{1}< t_{o}\) with \(P(t)\geq0\) for \(t\in[0,t_{1}]\) and \(P(t)<0\) for \(t\in(t_{1},t_{0})\). From the fact \(S(t)\geq 0\) and the first equation of (2.6) we have
$$\frac{dP(t)}{dt}\geq-b_{1}P(t)-a_{1}P^{2}(t)+mP(t), \quad t\in(t_{1},t_{0}). $$
Actually, the solution of \(\frac{dZ(t)}{dt}= -b_{1}Z(t)-a_{1}Z^{2}(t)+mZ(t)\) with \(Z(t_{1})= 0\) is the zero solution. Then the comparison principle leads to \(P(t)\geq0\) for \(t\geq t_{1}\), in contradiction with \(P(t)<0\), \(t\in(t_{1},t_{0})\), that is, \(P(t)\ge0\) if \(P(0)\geq0\), also on its maximal existence interval.
Note that the second equation in (2.5) can be rewritten as
$$\begin{aligned}& \frac{d}{dt}\bigl[I(t)e^{b_{3}t}\bigr] =\frac{d}{dt} \biggl[ \theta m \int_{t}^{t+\tau }\frac{P(\xi-\tau)S(\xi-\tau)}{S(\xi-\tau)+d}e^{b_{3}(\xi-\tau )}\,d\xi \biggr], \\& I(t)e^{b_{3}t} =I(0)+\theta m \int_{0}^{t}\biggl[ \int_{s}^{s+\tau}\frac{P(\xi -\tau)S(\xi-\tau)}{S(\xi-\tau)+d}e^{b_{3}(\xi-\tau)}\,d\xi \biggr] \,ds. \end{aligned}$$
Thus, the nonnegativity of \(P(t)\) and \(S(t)\) leads to the nonnegativity of \(I(t)\). Furthermore, \(I(t)\geq 0\) if \(I(0)\geq 0\).

If \(\phi_{1}(0)>0\) and \(\phi_{2}(0)>0\), then a similar argument leads to \((P(t),S(t))> (0,0)\) on its maximal existence interval. The proof is complete. □

The following lemma will lead to the global existence of solution \((P(t;\phi),S(t;\phi))\).

Lemma 3.2

(Boundedness) The solution \((P(t),S(t))^{T}\) of (2.6) satisfying the initial condition (3.1) is bounded as long as it exists.

Proof

We consider (2.5) with the initial conditions (3.1), where \((\phi_{1},\phi_{2},\phi_{3})^{T}\in C([\tau,0],\mathbb{R}_{+}^{3})\). In view of Lemma 3.1, we can obtain the existence of a unique local positive solution of (2.5), defined on its maximal interval \([0,T)\). Note that
$$\begin{aligned}& \theta\frac{dP(t)}{dt}+\frac{dI(t)}{dt}+\frac{dS(t)}{dt} \\& \quad = -\theta b_{1}P(t)-\theta a_{1}P^{2}(t)+ \theta m\frac{dP(t)}{S(t)+d} -b_{3}I(t)+\theta m\frac{P(t)S(t)}{S(t)+d} -b_{2}S(t)-a_{2}S^{2}(t) \\& \quad = \theta(m-b_{1})P(t)-\theta a_{1}P^{2}(t)-b_{3}I(t)-b_{2}S(t)-a_{2}S^{2}(t). \end{aligned}$$
Then, for any positive constant number \(\beta<\min\{b_{2},b_{3}\}\), by the positivity of solution we have
$$\begin{aligned}& \theta\frac{dP(t)}{dt}+\frac{dI(t)}{dt}+\frac{dS(t)}{dt}+\beta \bigl( \theta P(t)+I(t)+S(t) \bigr) \\& \quad \le \theta(\beta+m-b_{1})P(t)-\theta a_{1}P^{2}(t) \\& \quad = -\theta a_{1} \biggl(P(t)-\frac{\beta+m-b_{1}}{2a_{1}} \biggr)^{2}+\frac {\theta(\beta+m-b_{1})^{2}}{4a_{1}} \\& \quad \leq \frac{\theta(\beta+m-b_{1})^{2}}{4a_{1}}:=C_{0}. \end{aligned}$$
Furthermore, we have
$$\theta P(t)+S(t)+I(t)< \frac{C_{0}}{\beta}+ \biggl(\theta P(0)+S(0)+I(0)- \frac{C_{0}}{\beta} \biggr)e^{-\beta t}, $$
which implies that \((P(t),S(t))\) is bounded as long as it exists. □

In view of Lemmas 3.1 and 3.2, we can draw the following conclusion.

Theorem 3.1

There exists a positive solution of (2.6) defined on \([0,\infty)\) with the initial condition (3.1), which is unique. Furthermore, the solution semiflow \(\Phi(t)\) of (2.6), defined by \([\Phi(t)\phi](\xi)=(P_{t}(\xi),S_{t}(\xi))^{T}\), is well defined and has a global attractor in X.

4 Global dynamical properties

In this section, we look into the stability and Hopf bifurcation of system (2.6). Firstly, we investigate the existence of equilibria.

Lemma 4.1

The following conclusions for (2.6) hold.
  1. (1)

    The zero equilibrium (extinction equilibrium) \(E_{0}=(0,0)\) always exists.

     
  2. (2)

    A unique nonzero boundary equilibrium (scrounger-free equilibrium) \(E_{1}= (\frac{m-b_{1}}{a_{1}},0 ):=(p_{1},0)\) exists if and only if \(m>b_{1}\).

     
  3. (3)
    A unique positive equilibrium (coexistence equilibrium) \(E_{*}=(p^{*},s^{*})\) exists if and only if
    $$ \theta me^{-b_{3}\tau}(m-b_{1})>da_{1}b_{2}, $$
    where \(E_{*}=(p^{*},s^{*})\) satisfies
    $$-b_{1}-a_{1}p^{*}+m\frac{d}{s^{*}+d}=0, \qquad -b_{2}-a_{2}s^{*}+\theta m\frac {p^{*}}{s^{*}+d}e^{-b_{3}\tau}=0. $$
    In addition, if \(E_{*}\) exists, then \(E_{1}\) also exists.
     

Proof

The proof of (1) and (2) is obvious, so we omit it here. Now we prove (3). A coexistence equilibrium \(E_{*}\) exists if and only if the following algebraic system (4.1) has a pair of positive roots:
$$ -b_{1}-a_{1}p+m\frac{d}{s+d}=0,\qquad -b_{2}-a_{2}s+\theta m\frac {p}{s+d}e^{-b_{3}\tau}=0. $$
(4.1)
Solving (4.1) is equivalent to that two curves have a cross-point in the first quadrant of \(\mathbb{R}^{2}\). Note that the curve \(p=\frac{1}{a_{1}} (-b_{1}+m\frac{d}{s+d} )\) decreases as \(s\ge0\) increases and has a unique cross-point \((\frac {m-b_{1}}{a_{1}},0)\) with the p-axis. We know that \(p=\frac{e^{b_{3}\tau }}{\theta m}(b_{2}+a_{2}s)(s+d)\) increases to +∞ as s increases to +∞ and also has a unique cross-point \((\frac{e^{b_{3}\tau }}{\theta m}b_{2}d,0)\) with the p-axis. Thus, their p-intercepts are \(\frac{m-b_{1}}{a_{1}}\) and \(\frac{e^{b_{3}\tau}}{\theta m}b_{2}d\), respectively. Thus, a unique positive cross-point of the two curves exists if and only if their p-intercepts satisfy \(\frac {m-b_{1}}{a_{1}}>\frac{e^{b_{3}\tau}}{\theta m}b_{2}d\), that is, \(\theta me^{-b_{3}\tau}(m-b_{1})>da_{1}b_{2}\). The proof is complete. □

4.1 Basic reproduction ratio and stability of boundary equilibria

The following theorem describes the stability of extinction equilibrium \(E_{0}\).

Theorem 4.1

If \(m\le b_{1}\), then the extinction equilibrium \(E_{0}=(0,0)\) is globally asymptotically stable; if \(m>b_{1}\), then the extinction equilibrium \(E_{0}\) is unstable.

Proof

The linearized system of (2.6) at \(E_{0}\) is
$$ \textstyle\begin{cases} \frac{dP(t)}{dt}=-b_{1}P(t)+mP(t), \\ \frac{dS(t)}{dt}=-b_{2}S(t), \end{cases} $$
(4.2)
and the corresponding characteristic equation takes the form
$$ (\lambda-m+b_{1} ) (\lambda+b_{2} )=0. $$
(4.3)

It easy to see that (4.3) has and only has two roots, \(-b_{1}+m\) and \(-b_{2}\). If \(m>b_{1}\), then \(-b_{1}+m\) is positive. Thus, \(E_{0}\) is unstable if \(m>b_{1}\).

Now assume that \(m\le b_{1}\). Let
$$V_{1}(P,S) (t)=\theta e^{-b_{3}\tau}P(t)+S(t) +\theta me^{-b_{3}\tau} \int_{0}^{\tau}\frac{P(t-\eta)S(t-\eta )}{S(t-\eta)+d}\,d\eta. $$
Note that
$$\begin{aligned} \frac{d}{dt} \int_{0}^{\tau}\frac{P(t-\eta)S(t-\eta)}{S(t-\eta )+d}\,d\eta =& \int_{0}^{\tau}\frac{d}{dt}\frac{P(t-\eta)S(t-\eta)}{S(t-\eta )+d}\,d \eta \\ =&- \int_{0}^{\tau}\frac{d}{d\eta}\frac{P(t-\eta)S(t-\eta )}{S(t-\eta)+d}\,d \eta \\ =&-\frac{P(t-\tau)S(t-\tau)}{S(t-\tau)+d}+\frac{P(t)S(t)}{S(t)+d}. \end{aligned}$$
(4.4)
Then
$$\begin{aligned} \frac{dV_{1}(P,S)(t)}{dt} =&\theta e^{-b_{3}\tau}P'(t)+S'(t) +\theta me^{-b_{3}\tau} \biggl(\frac{P(t)S(t)}{S(t)+d}-\frac{P(t-\tau )S(t-\tau)}{S(t-\tau)+d} \biggr) \\ =& \theta e^{-b_{3}\tau} \biggl(-b_{1}P(t)-a_{1}P^{2}(t)+m \frac {dP(t)}{S(t)+d} \biggr) \\ &{}+ \biggl(-b_{2}S(t)-a_{2}S^{2}(t)+ \theta m\frac{P(t-\tau)S(t-\tau )}{S(t-\tau)+d}e^{-b_{3}\tau} \biggr) \\ &{}+\theta me^{-b_{3}\tau} \biggl(-\frac{P(t-\tau)S(t-\tau)}{S(t-\tau )+d}+\frac{P(t)S(t)}{S(t)+d} \biggr) \\ =& -a_{1}\theta e^{-b_{3}\tau}P^{2}(t)-b_{2}S(t)-a_{2}S^{2}(t)+ \theta e^{-b_{3}\tau}(m-b_{1})P(t). \end{aligned}$$
Since \(m\le b_{1}\), it follows that \(\frac{dV_{1}(P,S)(t)}{dt}\le0\), and \(\frac{dV_{1}(P,S)(t)}{dt}=0\) if and only if \((P,S)(t)\equiv (0,0)\). The proof is complete. □
Now, we define a basic reproduction ratio \(R_{0}\) for (2.6) (see [15]). Linearizing the second equation at the scrounger-free equilibrium \(E_{1}=(p_{1},0)\), we obtain
$$\frac{dS(t)}{dt}=\theta me^{-b_{3}\tau}\frac{p_{1}}{d}S(t-\tau )-b_{2}S(t)=\theta me^{-b_{3}\tau}\frac{p_{1}}{d}S_{t}(- \tau)-b_{2}S(t). $$
We then separate the last equation into two parts: \(\frac {dS(t)}{dt}=F(S_{t})-VS(t)\). The first part \(F(S_{t})\) describes the newly growth mature scroungers at time t depending linearly on the scrounging interaction over \([t-\tau,t]\), and the second part \(-VS(t)\) describes the internal evolution of mature scroungers. Here, we define the function F mapping \(C([-\tau,0],\mathbb{R})\) into \(\mathbb{R}\) by \(F(\phi)=\theta me^{-b_{3}\tau}\frac{p_{1}}{d}\phi(-\tau)\) and use \(VS(t)=b_{2}S(t)\). It is easy to see F is a nonnegative operator and the one-dimensional matrix −V is cooperative. Denote mapping \(\mathbb{R}\) in \(\mathbb{R}\) by \(\hat{F}u=F(\hat{u})\), where \(\hat{u}(\theta)\equiv u\) for \(\forall\theta\in[-\tau,0]\). Thus, \(\hat{F}=\theta me^{-b_{3}\tau}\frac{p_{1}}{d}\). Then, according to [15], Corollary 2.1, we have
$$R_{0}=r\bigl(\hat{F}V^{-1}\bigr)=\theta me^{-b_{3}\tau} \frac{p_{1}}{d}\cdot\frac{1}{b_{2}}, $$
where \(r(M)\) denotes the spectral radius of matrix M, and \(M^{-1}\) denotes the inverse matrix of M.

Remark 4.1

With \(R_{0}\) defined as before, we see that \(R_{0}>1\) is consistent with the condition for the existence of a unique coexistence equilibrium. Further, \(R_{0}\) is in proportion to the birth rate of producer without mature scrounger’s effect \((\theta m\frac{p_{1}}{d})\), the survival rate for immature scrounger (\(e^{-b_{3}\tau}\)), and the average lifetime of mature scrounger (\(\frac{1}{b_{2}}\)).

Theorem 4.2

Assume that \(m>b_{1}\). If \(R_{0}\le1\), then the scrounger-free equilibrium \(E_{1}=(p_{1},0)\) is globally asymptotically stable; if \(R_{0}>1\), then the scrounger-free equilibrium is unstable.

Proof

The linearized system of (2.6) at \(E_{1}\) is
$$ \textstyle\begin{cases} \frac{dP(t)}{dt}=-a_{1}p_{1}P(t)-m\frac{p_{1}}{d}S(t), \\ \frac{dS(t)}{dt}=-b_{2}S(t)+\theta m\frac{p_{1}S(t-\tau)}{d}e^{-b_{3}\tau}, \end{cases} $$
(4.5)
and the corresponding characteristic equation takes the form
$$ (\lambda+a_{1}p_{1} ) \biggl( \lambda+b_{2}-\theta m\frac {p_{1}}{d}e^{-b_{3}\tau}e^{-\lambda\tau} \biggr)=0. $$
(4.6)
Note that \(g(\lambda):=\lambda+b_{2}-\theta m\frac{p_{1}}{d}e^{-b_{3}\tau }e^{-\lambda\tau}\rightarrow\infty\) as \(\lambda\rightarrow\infty \). If \(R_{0}>1\), that is, \(g(0)=b_{2}-\theta m\frac{p_{1}}{d}e^{-b_{3}\tau }<0\), then we can confirm that \(g(\lambda)=0\) has a positive root. Thus, \(E_{1}\) is unstable.
Now assume that \(R_{0}\le1\). Let
$$V_{2}(P,S) (t)=\theta e^{-b_{3}\tau} \biggl(P(t)-p_{1}-p_{1} \ln\frac {P(t)}{p_{1}} \biggr)+S(t) +\theta me^{-b_{3}\tau} \int_{0}^{\tau}\frac{P(t-\eta)S(t-\eta )}{S(t-\eta)+d}\,d\eta. $$
We have from (4.4) that
$$\begin{aligned} \frac{dV_{2}(P,S)(t)}{dt} = &\theta e^{-b_{3}\tau}P'(t) \biggl(1- \frac{p_{1}}{P(t)} \biggr)+S'(t) \\ &{}+\theta me^{-b_{3}\tau} \biggl(- \frac{P(t-\tau)S(t-\tau)}{S(t-\tau )+d}+\frac{P(t)S(t)}{S(t)+d} \biggr) \\ =& \theta e^{-b_{3}\tau} \biggl(-b_{1}P(t)-a_{1}P^{2}(t)+m \frac {dP(t)}{S(t)+d} \biggr) \biggl(1-\frac{p_{1}}{P(t)} \biggr) \\ &{}+ \biggl(-b_{2}S(t)-a_{2}S^{2}(t)+\theta m \frac{P(t-\tau)S(t-\tau )}{S(t-\tau)+d}e^{-b_{3}\tau} \biggr) \\ &{}+\theta me^{-b_{3}\tau} \biggl(-\frac{P(t-\tau)S(t-\tau)}{S(t-\tau )+d}+\frac{P(t)S(t)}{S(t)+d} \biggr) \\ =& \theta e^{-b_{3}\tau} \biggl(-m+m\frac{d}{S(t)+d} \biggr) \bigl(P(t)-p_{1} \bigr)-a_{1}\theta e^{-b_{3}\tau} \bigl(P(t)-p_{1}\bigr)^{2} \\ &{} + \biggl(-b_{2}+\theta m\frac{P(t)}{S(t)+d}e^{-b_{3}\tau} \biggr)S(t)-a_{2}S^{2}(t) \\ =& -\theta me^{-b_{3}\tau}\frac{S(t) (P(t)-p_{1} )}{S(t)+d}-a_{1}\theta e^{-b_{3}\tau}\bigl(P(t)-p_{1}\bigr)^{2} \\ &{} +\theta me^{-b_{3}\tau} \biggl(-\frac{p_{1}}{R_{0}d}+\frac {p_{1}}{d}- \frac{p_{1}}{d}+\frac{P(t)}{S(t)+d} \biggr)S(t)-a_{2}S^{2}(t) \\ =& -\theta me^{-b_{3}\tau}\frac{S(t) (P(t)-p_{1} )}{S(t)+d}-a_{1}\theta e^{-b_{3}\tau}\bigl(P(t)-p_{1}\bigr)^{2}-a_{2}S^{2}(t) \\ &{}+\theta me^{-b_{3}\tau}\frac{p_{1}}{R_{0}d}(R_{0}-1)S(t) +\theta me^{-b_{3}\tau}\frac{(P(t)-p_{1})d-p_{1}S(t)}{d(S(t)+d)}S(t) \\ =& -a_{1}\theta e^{-b_{3}\tau}\bigl(P(t)-p_{1} \bigr)^{2}-a_{2}S^{2}(t) \\ &{}+\frac {\theta me^{-b_{3}\tau}p_{1}}{R_{0}d}(R_{0}-1)S(t) -\frac{\theta me^{-b_{3}\tau}p_{1}}{d(S(t)+d)}S^{2}(t). \end{aligned}$$
Since \(R_{0}\le1\), it follows that \(\frac{dV_{2}(P,S)(t)}{dt}\le0\), and \(\frac{dV_{2}(P,S)(t)}{dt}=0\) if and only if \((P,S)(t)\equiv (p_{1},0)\). The proof is complete. □

4.2 Persistence

Now we study the uniform persistence of system (2.6) (see [16]). Denote \(X_{0}=\{\phi=(\phi_{1},\phi_{2})\in X: \phi _{1}(0)>0,\phi_{2}(0)>0\}\) and \(\partial X_{0}=X\backslash X_{0}\). Obviously, \(X_{0}\) is an open set in X. Let \(\Phi(t)\phi=(P_{t},S_{t})(\phi)\) be the solution semiflow of (2.6) defined in Theorem 3.1. Furthermore, (2.6) and the boundedness of \((P(t),S(t))\) yield the boundedness of \((P^{\prime}(t),S^{\prime}(t))\) for \(t\geq0\). Now, by the Arzéla-Ascoli theorem we know that every positive half-orbit of Φ: \(\gamma^{+}{\phi}=\{\Phi(t)\phi: t\geq0\}\) is precompact. As usual, for every equilibrium E of (2.6), we call
$$W^{s}(E)=\Bigl\{ \phi\in X: \lim_{t\rightarrow\infty} \bigl\Vert \Phi (t)\phi-E\bigr\Vert =0\Bigr\} $$
‘the stable set of E’.

Theorem 4.3

Assume that \(R_{0}>1\). Then system (2.6) is uniformly persistent. That is, there exists a positive number ε such that \(\liminf_{t\rightarrow\infty}P(t)>\varepsilon\) and \(\liminf_{t\rightarrow\infty}S(t)>\varepsilon\).

Proof

Denote by \(\omega(\phi)\) the omega limit set of the positive half-orbit \(\gamma^{+}{\phi}=\{\Phi(t)\phi: t\ge0\}\) and \(X_{\partial}=\{\phi\in\partial X_{0}: \Phi(t)\phi\in\partial X_{0},t\ge0\}\).

For all \(\phi\in X_{\partial}\), we have \(\Phi(t)\phi\in\partial X_{0}=X\backslash X_{0}=\{\phi=(\phi_{1},\phi_{2})\in X: \phi_{1}(0)=0\mbox{ or }\phi_{2}(0)=0\}\) for any \(t\ge0\). This implies that either \(P(t)=0\) or \(S(t)=0\) for \(t\ge0\). If \(P(t)=0\) for some \(t_{0}\), then for \(t\ge0\), \(P(t)\equiv0\), and further we have \(\omega(\phi)=E_{0}\). Otherwise, \(P(t)>0\) and \(S(t)\equiv0\) for any \(t\ge0\), and then \(\omega(\phi )=E_{1}\). Thus, \(\bigcup_{\phi\in X_{\partial}}\omega(\phi)\subset M:=\{ E_{0},E_{1}\}\), and no subset of M forms a cycle in \(\partial X_{0}\).

We now prove that there exists \(\delta>0\) such that \(\limsup_{t\rightarrow\infty} S(t)\ge\delta\) for all solutions \((P(t),S(t))\). If it is false, then for any \(\delta>0\), there exists a solution \((P(t),S(t))\) such that \(\limsup_{t\rightarrow\infty} S(t)<\delta\). This leads to \(S(t)<\delta\) for sufficiently large t. By the first equation of (2.6), for sufficiently large t, we have
$$\frac{dP(t)}{dt}=-b_{1}P(t)-a_{1}P^{2}(t)+m \frac {dP(t)}{S(t)+d}>-b_{1}P(t)-a_{1}P^{2} (t)+m \frac{dP(t)}{\delta+d}. $$
Therefore, \(\liminf_{t\rightarrow\infty}P(t)\ge\frac{1}{a_{1}} (-b_{1}+m\frac{d}{\delta+d} )\) and \(P(t)\ge\frac{1}{a_{1}} (-b_{1}+m\frac{d}{\delta+d} )-\delta\) for sufficiently large t.
From the second equation of (2.6), for sufficiently large t, we have
$$\begin{aligned} \frac{dS(t)}{dt} =&-b_{2}S(t)-a_{2}S^{2}(t)+ \theta m\frac{P(t-\tau)S(t-\tau )}{S(t-\tau)+d}e^{-b_{3}\tau} \\ \ge&-b_{2}S(t)-a_{2}S^{2}(t)+\theta m \frac{ (\frac{1}{a_{1}} (-b_{1}+m\frac{d}{\delta+d} )-\delta )S(t-\tau)}{S(t-\tau )+d}e^{-b_{3}\tau}. \end{aligned}$$
Note the characteristic equation of
$$ \frac{dV(t)}{dt}=-b_{2}V(t)-a_{2}V^{2}(t)+ \theta m\frac{ (\frac {1}{a_{1}} (-b_{1}+m\frac{d}{\delta+d} )-\delta )V(t-\tau)}{V(t-\tau)+d}e^{-b_{3}\tau} $$
(4.7)
is
$$ \lambda=-b_{2}+\theta m\frac{ (\frac{1}{a_{1}} (-b_{1}+m\frac {d}{\delta+d} )-\delta )}{d}e^{-\lambda\tau}e^{-b_{3}\tau}. $$
(4.8)
In view of the assumption \(\theta me^{-b_{3}\tau }(m-b_{1})>da_{1}b_{2}\), we can choose δ small enough such that \(\theta me^{-b_{3}\tau} (\frac{m-b_{1}}{a_{1}}-\frac{m\delta }{a_{1}(\delta+d)}-\delta )>db_{2}\). This implies that (4.8) has a positive root and the solution \(V(t)\) of (4.7) grows exponentially. Furthermore, by comparison argument, \(S(t)\) can also grow exponentially, in contradiction with the boundedness of \(S(t)\). Therefore, \(\limsup_{t\rightarrow\infty} S(t)\ge\delta\) for all solutions \((P(t),S(t))\). This leads to
$$\limsup_{t\rightarrow\infty}\bigl\Vert \Phi(t)\phi-E_{0}\bigr\Vert \ge \delta, \qquad \limsup_{t\rightarrow\infty}\bigl\Vert \Phi(t)\phi -E_{1}\bigr\Vert \ge\delta. $$
For any \(\phi=(\phi_{1},\phi_{2})\in X\), define the generalized distance function d mapping X into \(\mathbb{R}_{+}\) by
$${d}(\phi)=\min\bigl\{ \phi_{1}(0),\phi_{2}(0)\bigr\} . $$
Then it follows that \(E_{0}\) and \(E_{1}\) are both isolated in X and
$$W^{s}(E_{0})\cap d^{-1}(0,\infty)=\emptyset,\qquad W^{s}(E_{1})\cap d^{-1}(0,\infty)=\emptyset, $$
where \(d^{-1}(0,\infty)=\{\phi\in X: d(\phi)>0\}\).

According to [16], Theorem 3, there exists a positive number ε such that \(\liminf_{t\rightarrow\infty }P(t)>\varepsilon\) and \(\liminf_{t\rightarrow\infty }S(t)>\varepsilon\). The proof is complete. □

4.3 Stability of coexistence equilibrium and Hopf bifurcation

In this section, we need some notation:
$$\begin{aligned}& u_{1}:=a_{1}p^{*}+a_{2}s^{*}+\theta m \frac{p^{*}}{s^{*}+d}e^{-b_{3}\tau},\qquad u_{0}:=a_{1}p^{*} \biggl(a_{2}s^{*}+\theta m\frac{p^{*}}{s^{*}+d}e^{-b_{3}\tau } \biggr), \\& v_{1}:=-\theta m\frac{dp^{*}}{(s^{*}+d)^{2}}e^{-b_{3}\tau},\qquad v_{0}:=-\theta m\frac{a_{1}d(p^{*})^{2}}{(s^{*}+d)^{2}}e^{-b_{3}\tau}+\theta m^{2}\frac{dp^{*}s^{*}}{(s^{*}+d)^{3}}e^{-b_{3}\tau}, \\& \omega_{0}:=\sqrt{\frac{-(u_{1}^{2}-v_{1}^{2}-2u_{0})+\sqrt {(u_{1}^{2}-v_{1}^{2}-2u_{0})^{2}-4(u_{0}^{2}-v_{0}^{2})}}{2}}, \\& \tau_{0}:=\frac{1}{\omega_{0}}\arccos{\frac{v_{0}(\omega _{0}^{2}-u_{0})-u_{1}v_{1}\omega_{0}^{2}}{v_{0}^{2}+v_{1}^{2}\omega_{0}^{2}}}. \end{aligned}$$
Our main results in this subsection are described in the following theorem concerning the stability of coexistence equilibrium and Hopf bifurcation.

Theorem 4.4

Assume that \(R_{0}>1\). Then we have the following conclusions.
  1. (1)

    The coexistence equilibrium \(E_{*}\) is globally asymptotic stable for \(\tau=0\).

     
  2. (2)

    If \(u_{0}>v_{0}\), then the coexistence equilibrium \(E_{*}\) is locally asymptotic stable for any \(\tau\ge0\).

     
  3. (3)

    If \(u_{0}< v_{0}\), then the coexistence equilibrium \(E_{*}\) is locally asymptotic stable for \(0\le\tau<\tau_{0}\) and unstable for \(\tau>\tau_{0}\). Furthermore, (2.6) undergoes a Hopf bifurcation at \(E_{*}\) when \(\tau=\tau_{0}\). Here \(u_{0}\), \(v_{0}\), \(\tau_{0}\) are constants defined previously.

     

Proof

The linearized system of (2.6) at \(E_{*}\) is
$$ \textstyle\begin{cases} \frac{dP(t)}{dt}=-b_{1}P(t)-2a_{1}p^{*}P(t)+m\frac{d P(t)}{s^{*}+d}-m\frac {dp^{*}S(t)}{(s^{*}+d)^{2}}, \\ \frac{dS(t)}{dt}=-b_{2}S(t)-2a_{2}s^{*}S(t)+\theta m\frac{s^{*}P(t-\tau )}{s^{*}+d}e^{-b_{3}\tau} +\theta m\frac{dp^{*} S(t-\tau)}{(s^{*}+d)^{2}}e^{-b_{3}\tau}, \end{cases} $$
(4.9)
and the characteristic equation of (4.9) takes the form
$$\begin{aligned}& \bigl(\lambda+a_{1}p^{*} \bigr) \biggl[ \lambda+a_{2}s^{*}+\theta m \biggl(\frac{p^{*}}{s^{*}+d} -\frac{dp^{*}e^{-\lambda\tau}}{(s^{*}+d)^{2}} \biggr)e^{-b_{3}\tau} \biggr] \\& \quad {}+\theta m^{2}\frac{dp^{*}s^{*}}{(s^{*}+d)^{3}}e^{-b_{3}\tau}e^{-\lambda\tau}=0. \end{aligned}$$
(4.10)
Case 1. If \(\tau=0\), then (4.10) becomes
$$\bigl(\lambda+a_{1}p^{*} \bigr) \biggl(\lambda+a_{2}s^{*}+ \theta m \frac{p^{*}s^{*}}{(s^{*}+d)^{2}} \biggr) +\theta m^{2}\frac{dp^{*}s^{*}}{(s^{*}+d)^{3}}=0. $$
It is easy to see this equation has exactly two negative roots, and thus the coexistence equilibrium \(E_{*}\) is locally asymptotic stable.
Define the Lyapunov function
$$V_{3}(P,S) (t)=\theta \biggl(P(t)-p^{*}-p^{*}\ln\frac{P(t)}{p^{*}} \biggr) +\frac{d}{s^{*}+d} \biggl(S(t)-s^{*}-s^{*}\ln\frac{S(t)}{s^{*}} \biggr). $$
Then the derivative along system (2.6) is
$$\begin{aligned} \frac{dV_{3}(P,S)(t)}{dt} =& \theta P'(t) \biggl(1-\frac{p^{*}}{P(t)} \biggr)+\frac{d}{s^{*}+d}S'(t) \biggl(1-\frac{s^{*}}{S(t)} \biggr) \\ =& \theta \biggl(-b_{1}P(t)-a_{1}P^{2}(t)+m \frac{dP(t)}{S(t)+d} \biggr) \biggl(1-\frac{p^{*}}{P(t)} \biggr) \\ &{} +\frac{d}{s^{*}+d} \biggl(-b_{2}S(t)-a_{2}S^{2}(t)+ \theta m\frac {P(t)S(t)}{S(t)+d} \biggr) \biggl(1-\frac{s^{*}}{S(t)} \biggr) \\ =& -\theta a_{1}\bigl(P(t)-p^{*}\bigr)^{2}- \frac{a_{2}d}{s^{*}+d}\bigl(S(t)-s^{*}\bigr)^{2} \\ &{}+\theta md \biggl( \frac{1}{S(t)+d}-\frac{1}{s^{*}+d} \biggr) \bigl(P(t)-p^{*}\bigr) \\ &{} -\frac{\theta md}{s^{*}+d} \biggl(\frac{p^{*}}{s^{*}+d}-\frac {P(t)}{S(t)+d} \biggr) \bigl(S(t)-s^{*}\bigr) \\ =& -\theta a_{1}\bigl(P(t)-p^{*}\bigr)^{2}- \frac{a_{2}d}{s^{*}+d}\bigl(S(t)-s^{*}\bigr)^{2} \\ &{}+\theta md\frac{s^{*}-S(t)}{(s^{*}+d)(S(t)+d)} \bigl(P(t)-p^{*}\bigr) \\ &{} -\frac{\theta md}{s^{*}+d}\frac {p^{*}(S(t)-s^{*})+(s^{*}+d)(p^{*}-P(t))}{(s^{*}+d)(S(t)+d)}\bigl(S(t)-s^{*}\bigr) \\ =& -\theta a_{1}\bigl(P(t)-p^{*}\bigr)^{2}- \frac{a_{2}d}{s^{*}+d}\bigl(S(t)-s^{*}\bigr)^{2}-\frac{\theta md}{s^{*}+d} \frac{p^{*}(S(t)-s^{*})^{2}}{(s^{*}+d)(S(t)+d)}\le0. \end{aligned}$$
Furthermore, \(\frac{dV_{3}(P,S)(t)}{dt}=0\) if and only if \((P,S)(t)\equiv(p^{*},s^{*})\). Thus, \(E_{*}\) is globally asymptotically stable when \(\tau=0\).
Case 2. If \(\tau>0\), denote
$$ u(\lambda)=\lambda^{2}+u_{1}\lambda+u_{0},\qquad v(\lambda)=v_{1}\lambda+v_{0}. $$
Then (4.10) can be rewritten as \(g(\lambda,\tau)=0\), where
$$g(\lambda,\tau):=u(\lambda)+v(\lambda)e^{-\lambda\tau}=\bigl(\lambda ^{2}+u_{1}\lambda+u_{0}\bigr)+(v_{1} \lambda+v_{0})e^{-\lambda\tau}. $$
By calculation we have
$$\begin{aligned}& u_{1}^{2}-v_{1}^{2}-2u_{0} \\& \quad = \bigl(a_{1}p^{*}\bigr)^{2}+ \biggl(a_{2}s^{*}+ \theta m\frac{p^{*}}{s^{*}+d}e^{-b_{3}\tau } \biggr)^{2} - \biggl(\theta m \frac{dp^{*}}{(s^{*}+d)^{2}}e^{-b_{3}\tau} \biggr)^{2} \\& \quad = \bigl(a_{1}p^{*}\bigr)^{2}+ \biggl(a_{2}s^{*}+ \theta m\frac{p^{*}}{s^{*}+d}e^{-b_{3}\tau }+\theta m\frac{dp^{*}}{(s^{*}+d)^{2}}e^{-b_{3}\tau} \biggr) \biggl(a_{2}s^{*}+\theta m\frac{p^{*}s^{*}}{(s^{*}+d)^{2}}e^{-b_{3}\tau} \biggr) \\& \quad >0 \end{aligned}$$
and
$$\begin{aligned} u_{0}+v_{0} =&a_{1}a_{2}p^{*}s^{*}+\theta m\frac{a_{1}(p^{*})^{2}}{s^{*}+d}e^{-b_{3}\tau} -\theta m\frac{a_{1}d(p^{*})^{2}}{(s^{*}+d)^{2}}e^{-b_{3}\tau}+ \theta m^{2}\frac {dp^{*}s^{*}}{(s^{*}+d)^{3}}e^{-b_{3}\tau} \\ =&a_{1}a_{2}p^{*}s^{*} +\theta m\frac{a_{1}s^{*}(p^{*})^{2}}{(s^{*}+d)^{2}}e^{-b_{3}\tau}+ \theta m^{2}\frac {dp^{*}s^{*}}{(s^{*}+d)^{3}}e^{-b_{3}\tau} \\ >&0. \end{aligned}$$
This leads to that the sign of \(u_{0}^{2}-v_{0}^{2}\) is consistent with the sign of \(u_{0}-v_{0}\).
Assuming that \(\pm i\omega\) is a pair of pure imaginary roots of (4.10) such that
$$g(i\omega,\tau)=u(i\omega)+v(i\omega)e^{-i\omega\tau}=0, $$
which yields \(u(i\omega)=-v(i\omega)e^{-i\omega\tau}\), and thus \(|u(i\omega)|^{2}-|v(i\omega)|^{2}=0\). Now there exist two subcases, \(u_{0}>v_{0}\) and \(u_{0}< v_{0}\).
Subcase 2a. If \(u_{0}>v_{0}\), according to [17] (also see [18], Theorem 4.1),
$$G(\omega):=\bigl\vert u(i\omega)\bigr\vert ^{2}-\bigl\vert v(i \omega)\bigr\vert ^{2}=\omega ^{4}+\bigl(u_{1}^{2}-v_{1}^{2}-2u_{0} \bigr)\omega^{2}+u_{0}^{2}-v_{0}^{2} $$
has no positive roots, and thus no stability switch occurs, that is, the coexistence equilibrium \(E_{*}\) is locally asymptotic stable for any \(\tau\ge0\) from its stability for \(\tau=0\).
Subcase 2b. If \(u_{0}< v_{0}\), \(G(\omega)\) has a unique simple positive root
$$\omega_{0}=\sqrt{\frac{-(u_{1}^{2}-v_{1}^{2}-2u_{0})+\sqrt {(u_{1}^{2}-v_{1}^{2}-2u_{0})^{2}-4(u_{0}^{2}-v_{0}^{2})}}{2}}, $$
and as τ increases, stability switch occurs. Eventually the coexistence equilibrium \(E_{*}\) becomes unstable. In detail, \(\pm i\omega _{0}\) is a pair of pure imaginary roots of (4.10) such that
$$g(i\omega_{0},\tau)=\bigl(-\omega_{0}^{2}+iu_{1} \omega_{0}+u_{0}\bigr)+(iv_{1}\omega _{0}+v_{0})e^{-i\omega_{0}\tau}=0. $$
Separating its real and imaginary parts, we have
$$\begin{aligned}& u_{0}-\omega_{0}^{2}+v_{1} \omega_{0}\sin{\omega_{0}\tau}+v_{0}\cos{\omega _{0}\tau}=0, \\& u_{1}\omega_{0}+v_{1} \omega_{0}\cos{\omega_{0}\tau}-v_{0}\sin { \omega_{0}\tau}=0. \end{aligned}$$
Thus we obtain the following critical values for delay τ:
$$\tau_{k}=\frac{1}{\omega_{0}}\arccos{\frac{v_{0}(\omega _{0}^{2}-u_{0})-u_{1}v_{1}\omega_{0}^{2}}{v_{0}^{2}+v_{1}^{2}\omega_{0}^{2}}}+ \frac{2k\pi }{\omega_{0}},\quad k=0,1,2,\ldots. $$
By the differentiation method of implicit functions we obtain from \(g(\lambda,\tau)=0\) that
$$\begin{aligned} \begin{aligned} \frac{d\tau}{d\lambda}&= \biggl(\frac{d\lambda}{d\tau} \biggr)^{-1} \\ &=\frac{2\lambda+u_{1}}{\lambda(v_{1}\lambda+v_{0})e^{-\lambda\tau }}+\frac{v_{1}}{\lambda(v_{1}\lambda+v_{0})}-\frac{\tau}{\lambda} \\ &=-\frac{2\lambda+u_{1}}{\lambda(\lambda^{2}+u_{1}\lambda+u_{0})}+\frac {v_{1}}{\lambda(v_{1}\lambda+v_{0})}-\frac{\tau}{\lambda}. \end{aligned} \end{aligned}$$
Note that
$$G(\omega_{0})=\omega_{0}^{4}+ \bigl(u_{1}^{2}-v_{1}^{2}-2u_{0} \bigr)\omega _{0}^{2}+u_{0}^{2}-v_{0}^{2}=u_{1}^{2} \omega_{0}^{2}+\bigl(u_{0}-\omega_{0}^{2} \bigr)^{2}-\bigl(v_{1}^{2}\omega _{0}^{2}+v_{0}^{2}\bigr)=0, $$
and then we have
$$\begin{aligned} \operatorname{Re} \biggl(\frac{d\lambda}{d\tau} \biggr)^{-1} \bigg|_{\tau =\tau_{k},\lambda=i\omega_{0}} &=\operatorname{Re} \biggl(-\frac{2i\omega_{0}+u_{1}}{i\omega_{0}(-\omega _{0}^{2}+i\omega_{0}u_{1}+u_{0})}+ \frac{v_{1}}{i\omega_{0}(i\omega _{0}v_{1}+v_{0})} \biggr) \\ &=-\frac{-u_{1}^{2}+2(u_{0}-w_{0}^{2})}{u_{1}^{2}\omega_{0}^{2}+(u_{0}-\omega _{0}^{2})^{2}}-\frac{v_{1}^{2}}{v_{1}^{2}\omega_{0}^{2}+v_{0}^{2}} \\ &=\frac{(2\omega_{0}^{2}+u_{1}^{2}-v_{1}^{2}-2u_{0})(v_{1}^{2}\omega _{0}^{2}+v_{0}^{2})-v_{1}^{2}G(\omega_{0})}{[u_{1}^{2}\omega_{0}^{2}+(u_{0}-\omega _{0}^{2})^{2}](v_{1}^{2}\omega_{0}^{2}+v_{0}^{2})} \\ &=\frac{2\omega_{0}^{2}+u_{1}^{2}-v_{1}^{2}-2u_{0}}{u_{1}^{2}\omega_{0}^{2}+(u_{0}-\omega_{0}^{2})^{2}} =\frac{\sqrt{(u_{1}^{2}-v_{1}^{2}-2u_{0})^{2}-4(u_{0}^{2}-v_{0}^{2})}}{u_{1}^{2}\omega _{0}^{2}+(u_{0}-\omega_{0}^{2})^{2}}>0. \end{aligned}$$
Combining the stability of \(E_{*}\) for \(\tau=0\), this implies that the transversal condition for Hopf bifurcation holds at \(\tau=\tau_{0}\). The proof is complete. □

In fact, we could discuss the direction and stability of the Hopf bifurcation by using the normal form theory and the center manifold reduction introduced by Hassard et al. [19]. However, the algorithm is standard, and the application of the result needs complex computations. Therefore, we just do some numerical simulations in the next section instead.

5 Numerical simulations and concluding discussions

We now give numerical simulations as \(R_{0}>1\) for (2.6) with some different values of parameters (see Figures 1-5). We can see from Figures 4 and 5 that the dynamical behavior would be very complex when \(R_{0}>1\) and \(u_{0}< v_{0}\) with large τ. That is, the time delay τ has really a huge impact on the dynamical behavior of (2.6).
Figure 1

The case of \(\pmb{R_{0}>1}\) and \(\pmb{u_{0}>v_{0}}\) , where \(\pmb{b_{1}=1}\) , \(\pmb{a_{1}=1}\) , \(\pmb{b_{2}=0.8}\) , \(\pmb{a_{2}=2}\) , \(\pmb{m=3}\) , \(\pmb{d=1}\) , \(\pmb{\theta e^{-b_{3}\tau}=3}\) , \(\pmb{\tau =0.5}\) .

Figure 2

The case of \(\pmb{R_{0}>1}\) and \(\pmb{u_{0}=60.3465}\) , \(\pmb{v_{0}=112.4246}\) , \(\pmb{\tau _{0}=0.6461}\) , where \(\pmb{b_{1}=10}\) , \(\pmb{a_{1}=1}\) , \(\pmb{b_{2}=0.8}\) , \(\pmb{a_{2}=10}\) , \(\pmb{m=30}\) , \(\pmb{d=1}\) , \(\pmb{\theta=3}\) , \(\pmb{\tau=0.5}\) , \(\pmb{b_{3}=2}\) , \(\pmb{P(t)=1}\) , \(\pmb{S(t)=1.65}\) for \(\pmb{t\in[-0.5,0]}\) .

Figure 3

The case of \(\pmb{R_{0}>1}\) and \(\pmb{u_{0}=44.0951}\) , \(\pmb{v_{0}=120.0637}\) , \(\pmb{\tau _{0}=0.5223}\) , where \(\pmb{b_{1}=10}\) , \(\pmb{a_{1}=1}\) , \(\pmb{b_{2}=0.8}\) , \(\pmb{a_{2}=10}\) , \(\pmb{m=30}\) , \(\pmb{d=1}\) , \(\pmb{\theta=3}\) , \(\pmb{\tau=0.55}\) , \(\pmb{b_{3}=1}\) , \(\pmb{P(t)=0.9}\) , \(\pmb{S(t)=1.7}\) for \(\pmb{t\in[-0.55,0]}\) .

Figure 4

The case of \(\pmb{R_{0}>1}\) and \(\pmb{u_{0}=81.8961}\) , \(\pmb{v_{0}=101.5272}\) , \(\pmb{\tau _{0}=1.1423}\) , where \(\pmb{b_{1}=10}\) , \(\pmb{a_{1}=1}\) , \(\pmb{b_{2}=0.8}\) , \(\pmb{a_{2}=10}\) , \(\pmb{m=30}\) , \(\pmb{d=1}\) , \(\pmb{\theta=3}\) , \(\pmb{\tau=3}\) , \(\pmb{b_{3}=0.5}\) , \(\pmb{P(t)=2}\) , \(\pmb{S(t)=1.5}\) for \(\pmb{t\in[-3,0]}\) .

Figure 5

The case of \(\pmb{R_{0}>1}\) and \(\pmb{u_{0}=50.9158}\) , \(\pmb{v_{0}=116.9143}\) , \(\pmb{\tau _{0}=0.5647}\) , where \(\pmb{b_{1}=10}\) , \(\pmb{a_{1}=1}\) , \(\pmb{b_{2}=0.8}\) , \(\pmb{a_{2}=10}\) , \(\pmb{m=30}\) , \(\pmb{d=1}\) , \(\pmb{\theta=3}\) , \(\pmb{\tau=3}\) , \(\pmb{b_{3}=0.25}\) , \(\pmb{P(t)=2}\) , \(\pmb{S(t)=1.5}\) for \(\pmb{t\in[-3,0]}\) .

In the following, we give some discussion on the influence of maturation period τ in the scrounge interaction.

The existence of producer. The existence of producer is just determined by their ability to discover food (m) and the natural mortality rate \(b_{1}\). However, the mature time τ does not determine the existence of producer.

The existence of scrounger. The basic reproduction ratio \(R_{0}\) determines the existence of scrounger. Note that \(R_{0}=\theta m\frac{p_{1}}{db_{2}}e^{-b_{3}\tau}\) decreases as τ increases, and it is a bounded function of variable τ. We find that (i) if \(\theta m p_{1}< db_{2}\), no matter how long the mature time is, the scrounger finally extincts; (ii) if \(\theta m p_{1}>db_{2}\), shorter mature time is in favor of the existence of scrounger.

Global behaviors for \(R_{0}>1\). In the case \(R_{0}>1\), we have no results of the global stability for \(\tau>0\). But in view of the global convergence to the coexistence equilibrium \(E_{*}\) at \(\tau=0\) and numerical simulations, we found that the mature time τ does change the dynamical behavior of the model. We conjecture that (i) when \(u_{0}>v_{0}\), the coexistence equilibrium is globally asymptotically stable for \(\tau>0\); (ii) when \(u_{0}< v_{0}\), the coexistence equilibrium remains globally asymptotically stable for small \(\tau>0\); (iii) when \(u_{0}< v_{0}\), the positive solutions convergence to a periodic solution for large τ.

Note that we also derived another model (2.7) in Section 2, whereas there exists competition between immature scroungers. The investigation of this model seems challenging. We leave it as an open problem.

We want to make some more comments on open problems. In this article and [4], we both assume that the proportion of the resource ultimately acquired by producer after the interaction is \(\frac {d}{S+d}\), where S is the number of scroungers. However, there may be some other models for different species groups by using different proportion functions. In this meaning, we can consider the following more general model for (1.1) with homogenous spatial environment:
$$ \textstyle\begin{cases} \frac{dP(t)}{dt}=-b_{1}P(t)-a_{1}P^{2}(t)+mg(P,S,d)P(t), \\ \frac{dS(t)}{dt}-b_{2}S(t)-a_{2}S^{2}(t)+\theta m(1-g(P,S,d))P(t). \end{cases} $$
(5.1)
If \(g(P,S,d)\equiv1\), that is, the scrounger is incapable of stealing food from producer, (5.1) becomes
$$ \textstyle\begin{cases} \frac{dP(t)}{dt}=(m-b_{1})P(t)-a_{1}P^{2}(t), \\ \frac{dS(t)}{dt}-b_{2}S(t)-a_{2}S^{2}(t). \end{cases} $$
(5.2)

If \(g(P,S,d)=\frac{d}{S+d}\), (5.1) becomes (1.1) with homogenous spatial environment.

We can also consider other cases such as \(g(P,S,d)=\frac{d^{2}}{S+d^{2}}\) for different species and such that even g actually depends on the producer P. Furthermore, one can derive a new model that involves a continuous immature age structure. We leave these problems for future research.

Declarations

Acknowledgements

Research is supported by NSF of China (11171120) and the Natural Science Foundation of Guangdong Province (2016A030313426). We are very grateful to the anonymous referee and the editors for a careful reading and helpful suggestions, which led to an improvement of our original manuscript.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Mathematics, South China Normal University

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