OSCILLATIONS IN AGE-STRUCTURED MODELS OF CONSUMER-RESOURCE MUTUALISMS Modeling consumer-resource interactions and understanding the

. In consumer-resource interactions, a resource is regarded as a biotic population that helps to maintain the population growth of its consumer, whereas a consumer exploits a resource and then reduces its growth rate. Bi-directional consumer-resource interactions describe the cases where each species acts as both a consumer and a resource of the other, which is the basis of many mutualisms. In uni-directional consumer-resource interactions one species acts as a consumer and the other as a material and/or energy resource while neither acts as both. In this paper we consider an age-structured model for uni-directional consumer-resource mutualisms in which the consumer species has both positive and negative eﬀects on the resource species, while the resource has only a positive eﬀect on the consumer. Examples include a predator-prey system in which the prey is able to kill or consume predator eggs or larvae and the insect pollinator and the host plant relationship in which the plants provide food, seeds, nectar and other resources for the pollinators while the pollinators have both positive and negative eﬀects on the plants. By carrying out local analysis and bifurcation analysis of the model, we discuss the stability of the positive equilibrium and show that under some conditions a non-trivial periodic solution through Hopf bifurcation appears when the mat- uration parameter passes through some critical values.


1.
Introduction. Consumer-resource interactions are closely related the process of energy and/or nutrient transfer between a consumer organism and a resource. Here a resource is regarded as a biotic population that helps to maintain the population growth of its consumer, whereas a consumer exploits a resource and then reduces its growth rate. Modeling consumer-resource interactions and understanding the nonlinear dynamics of such interactions has been one of the most important and active topics in ecology in the last four decades (MacArthur [13], Murdoch et al. [20]). Traditionally a consumer-resource interaction is modeled by using (+ −) (predation, parasitism) type relation in which the consumer gains some material benefit at the cost of the resource, such as the classical predator-prey or parasitehost models (Rosenzweig and MarArthur [22], May [18]).
Recently, mutualism has been studied explicitly in terms of consumer-resource interactions, such as (+ 0) (commensalism), (− 0) (amensalism), and (+ +) (mutualism), based on the balance between benefit and cost for the interacting species. For example, a mutualistic consumer exploits a resource (nutrient or nectar) supplied by another mutualistic species so that both the consumer and resource benefit from their interaction, which is described by a (+ +) type relation. Such mutualisms tend to be bi-directional, including coral mutualisms and mycorrhizal mutualisms (Holland and DeAngelis [7,8]), in which each species acts as both a consumer and a resource of the other. For instance, the coral polyp passes nitrogen from captured prey to the photosynthetic zooxanthellae while the zooxanthellae provide energy in the form of glucose to the coral animals. Terrestrial plants and mycorrhizal fungi in the rhizosphere of the root system have a mutualistic relationship (Wang et al. [27]).
The uni-directional consumer-resource mutualisms are consistent with the traditional consumer-resource interaction, in which one species acts as a consumer and the other as a material and/or energy resource, while neither acts as both. Resources produced by a mutualistic species (N 1 ) attract and reward a consumer (N 2 ), which in the process of exploring the resource provinsions N 1 with a service of dispersal or defense (Holland and DeAngelis [7,8], Wang et al. [27]). By assuming that the consumer species is age-structured, we consider the following consumerresource interaction model coupled by an ordinary differential equation (ODE) and a partial differential equation (PDE) where N 1 (t) represents the density of the resource species at time t and N 2 (t, a) represents the density of the consumer species at time t with age a. The number r is the intrinsic growth rate of the resource species and d 1 > 0 represents a logistic type limitation of the resource species (i.e. limitation for space, foods, etc.) so that r/d 1 > 0 is its carrying capacity when in isolation from the consumer. The function β(a) is the age-dependent maturation function so that is the the number of matured (reproducing) consumers. The term α 12 N 1 (t)A(t) γ 2 + A(t) describes the positive feedback on the growth of the resource species N 1 due to mutualistic interactions with the consumer species N 2 , where α 12 denotes the saturation level of the functional response of the consumer species and γ 2 denotes the half-saturation density of resource species; β 1 N 1 (t)A(t) represents the consumption level of resource species by matured consumer species. The number d 2 denotes the death rate of the consumer species. The term in the boundary condition denotes the new population births of the consumer species N 2 depending on resource supplied by N 1 , which saturates with resource density (N 1 ) according to an Michaelis-Menton function, where α 21 is the interaction strength and γ 1 is the half-saturation constant. System (1) is a generalization of the ODE model (2.1) of Wang and DeAngelis [26] on uni-directional consumerresource interactions. As pointed out by Wang et al. [27], such interactions may be modeled by age-structured models. This is the motivation of this article. Moreover, Wang and DeAngelis [26] showed that there is no periodic orbit in their ODE model and all solutions converge to a steady state. We will show that under some conditions a non-trivial periodic solution of the age-structured model (1) appears through a Hopf bifurcation when the maturation parameter passes through some critical values.
The insect pollinator and the host plant relationship is an example of the unidirectional consumer-resource mutualisms as the insect provides no material resource to the plant (though it provides a pollination service), see Holland and DeAngelis [7]. Pollinators travel from their nest to a foraging patch, collecting food, flying back to their nests, and unloading food. Interacting with flowers individually, the pollinators remove nectar, contact pollen, and provide pollination service. Therefore, the plants provide food, seeds, nectar and other resources for the pollinators, while the pollinators have both positive and negative effects on the plants. The positive effect of pollinators on plants is described by the Michaelis-Meton functional response α 12 N 1 (t)A(t)/(γ 2 + A(t)), where the parameter α 12 is regarded as the plants' efficiency in translating plant-pollinator interactions into fitness and α 21 is the corresponding value for the pollinators; β 1 denotes the percapita negative effect of pollinators on plants (Holland and DeAngelis [7], Wang, DeAngelis and Holland [28], and Mitchell et al. [19]).
Another example of consumer-resource interaction is introduced by Barkai and McQuaid [1] where they consider in some South African islands, rock lobsters feed on whelks, but in other areas whelks may be in such high abundance that they overwhelm and consume the lobsters. Also, Magalhães et al. [17] observed that small juvenile predatory mites may be killed by their thrips prey. Polis et al. [21] noted that 90 species of jellyfish and ctenophores eat fish eggs or larvae, while the older fish feed on these same species.
Before presenting our analysis and simulations of model (1), we make the following assumption. where τ ≥ 0, β * > 0 and 0 < R 0 < +∞. Assumption 1.1 indicates that there is a maturation period τ > 0, so that the maturation rate of the consumer species is β * > 0 when the age a is less than τ and zero when the age a is greater than τ. We will use the maturation period τ as the bifurcation parameter to study the stability of the positive equilibrium and the existence of a Hopf bifurcation in the age-structured model (1).
The rest of the paper is organized as follows. in next section we recall the general Hopf bifurcation theorem for the semilinear Cauchy problem with a non-densely defined domain. Section 3 deals with the stability of the positive steady state and existence of Hopf bifurcation in the age-structured consumer-resource model (1). Some numerical simulations and a brief discussion are given in section 4.

Hopf bifurcation theorem for nondensely defined Cauchy problems.
For convenience, we recall the general Hopf bifurcation theorem we established in Liu et al. [11]. Consider the semilinear Cauchy problem: where µ ∈ R is the bifurcation parameter, A : D(A) ⊂ X → X is a linear operator on a Banach space X with D(A) not dense in X and A not necessary a Hille-Yosida operator, F : We denote by {T A (t)} t≥0 the strongly continuous semigroup of bounded linear operators on X (respectively {S A (t)} t≥0 the integrated semigroup) generated by A. The essential growth bound ω 0,ess (L) ∈ (−∞, +∞) of L is defined by We make the following assumptions on the linear operator A and the nonlinear map F .
Banach space (X, · ) such that there exist two constants ω A ∈ R and M A ≥ 1, such that (ω A , +∞) ⊂ ρ(A) and the following properties are satisfied Assume that the following conditions are satisfied (iii) The essential growth bound of {T A0 (t)} t≥0 is strictly negative, that is, Now we can state the Hopf bifurcation theorem obtained in Liu et al. [11].
, which is an integrated solution of (3) with the parameter value equals µ(ε) and the initial value equals x ε . So for each t ≥ 0, u ε satisfies Moreover, we have the following properties (i) There exist a neighborhood N of 0 in X 0 and an open interval I in R containing 0, such that for µ ∈ I and any periodic solution u(t) in N with minimal period γ close to 2π ω(0) of (3) for the parameter value µ, there exists ε ∈ (0, ε * ) such that u(t) = u ε (t + θ) (for some θ ∈ [0, γ (ε))), µ(ε) = µ, and γ (ε) = γ.
(ii) The map ε → µ(ε) is a C k−1 function and we have the Taylor expansion where ω(0) is the imaginary part of λ (0) defined in Assumption 2.3.

3.
Equilibrium stability and Hopf bifurcation. In this section we investigate the stability and Hopf bifurcation of the age-structured consumer-resource model (1).
3.1. Rescaling time and age. In order to use the parameter τ as a bifurcation parameter (i.e. in order to obtain a smooth dependency of the system (1) with respect to τ ) we first normalize τ in (1) by the time-scaling and age-scaling a = a τ and t = t τ and consider the following distribution By dropping the hat notation we obtain, after this change of variable, the new system , with the new function β(a) defined by where τ ≥ 0, β * > 0 and 0 < R 0 < +∞.

3.2.
The transformation of the Cauchy problem. Consider the Banach space Let δ > 0 be fixed. Define the linear operator L : D(L) → X by Notice that L is non-densely defined since Let F : D(L) → X be the nonlinear operator defined by Then by setting we can rewrite system (5) as the following non-densely defined abstract Cauchy problem The global existence and uniqueness of solutions of system (7) follow from the results of Magal [14] and Magal and Ruan [15].
By solving the above equations, we obtain the following lemma.
Lemma 3.1. The system (7) always has the equilibria

ZHIHUA LIU, PIERRE MAGAL AND SHIGUI RUAN
Furthermore, there exists a unique positive equilibrium of system (7) x(a) = if and only if

The characteristic equation.
In order to get the linearized equation around the positive equilibrium x(a), we make the following change of variable We obtain Therefore the linearized equation of (8) around the equilibrium 0 is given by Then (8) can be written as where is a linear operator and satisfying H(0) = 0 and DH(0) = 0. Let υ := min{δ, τ d 2 }.
Then A = L + DF (x) = L + B By applying the results of Liu et al. [11], we obtain the following result. Moreover, L is a Hille-Yosida operator and satisfies Define the part of L in D(L) by L 0 , Thus we have ω 0,ess ( L 0 ) ≤ ω 0 ( L 0 ) ≤ −υ.
Hence we obtain the following proposition. Consider
Thus we have i.e.
This completes the proof.
From the above discussion about g(ζ) = 0, we know that for any k ∈ N 0 , there exists τ k such that the characteristic equation has two simple complex roots λ(τ ) = τ ζ(τ ) = τ α(τ ) ± iτ ω(τ ) that cross the imaginary axis transversely at τ = τ k : Summarizing the above results, we obtain the following conclusion.
By Theorem 2.4, the above results can be summarized as the following Hopf bifurcation theorem for system (1).
We would like to mention that the stability of the bifurcated periodic solutions can be determined by using the normal form theory developed in our recent work Liu et al. [12]. 4. Numerical simulations and discussions. Recently, Wang and DeAngelis [26] considered a specific uni-directional consumer-resource mutualism model in which the consumer species has both positive and negative effects on the resource species, while the resource has only a positive effect on the consumer, such as a predator-prey system in which the prey is able to kill or consume predator eggs or larvae.
In this article we generalized the ODE model of (2.1) of Wang and DeAngelis [26] to an age-structured model coupled by an ODE and a PDE, which describes uni-directional consumer-resource mutualism interactions with one species acting as a consumer and the other as a material and/or energy resource. Examples of such uni-directional consumer-resource mutualisms include the predator-prey systems in which the prey is able to kill or consume predator eggs or larvae, and the insect pollinator and the host plant relationship in which the plants provide food, seeds, nectar and other resources for the pollinators while the pollinators have both positive and negative effects on the plants. By carrying out local analysis and bifurcation analysis of the model, we discussed the stability of the positive equilibrium and found that under some conditions a non-trivial periodic solution through Hopf bifurcation appears when the maturation period of the consumer species τ passes through critical values τ = τ k .
In the following, we provide some numerical simulations to illustrate the stability of the positive equilibrium and the existence of a Hopf bifurcation for system (1). Choose parameters r = 4, α 21 = α 12 = β 1 = d 1 = 0.5, d 2 = 1.0, γ 1 = γ 2 = 0.5, and With these parameter values, we obtain numerically that τ 0 is approximately equal to 12.55. Under the same initial values we choose τ = 10 in Figure 1 and τ = 50 in Figure 2, respectively, and obtain graphs N 1 (t) and N 2 (t, a) by using Matlab. Figure 1 and 2 demonstrate that the positive equilibrium (N 1 , N 2 ) of system (1) is asymptotically stable when the maturation period is less than its first critical value and system (1) undergoes a Hopf bifurcation and a non-trivial periodic solution bifurcates from the positive equilibrium when the maturation period passes through the critical value. Notice that the ordinary differential equation version of model (1) does not exhibit oscillatory behavior (Wang and DeAngelis [26]). It is well-known that periodic oscillations via limit cycles are common in predator-prey systems (May [18]). The existence of periodic solutions in system (1) via bifurcation demonstrates that the age-structured model has more dynamic possibilities than the unstructured model. It is shown that both consume and resource species are more likely to coexist in oscillatory modes when the maturation period of the consumer species is long enough.
It has been observed that Hopf bifurcation occurs in age-structured models (see Cushing [3], Magal and Ruan [16], and the references cited therein). Recently, by rewriting age-structured systems as nondensely defined Cauchy problems, we established a Hopf bifurcation theorem for a general class of age-structured models (Liu et al. [11]). Due to the complexity of analysis and computations, applications of this general Hopf bifurcation theorem mainly focus on single species age-structured models. In this article we applied the techniques and results to a uni-directional consume-resource mutualism model coupled of one ordinary differential equation   and one age-structured equation. We would like to point out that, due to the form of the age-dependent maturation function β(a), system (1) could be handled by reducing it to a system of delay differential equations. Nevertheless, we would like to use our recent results and techniques to treat this model in the age-structured model setting and believe that similar results hold for more general forms of agedependent maturation functions. Moreover, such a model structure is similar to the classical predator-prey interaction systems, but is different. The nonlinear dynamics of age-structured predator-prey population models have been studied by many researchers, see for example, Cushing [2,3], Cushing and Saleem [4], Gurtin and Levine [6], Levine [9], Li [10], Saleem [23], and Venturino [25], and various interesting asymptotical behaviors including bifurcation have been observed. It will be very interesting to apply the general Hopf bifurcation theorem in Liu et al. [11] to study Hopf bifurcations in predator-prey population models when both predator and prey species are age-structured.