Existence and uniqueness for a class of nonlinear population diffusion system

ABSTRACT In this paper, we introduce a class of nonlinear population diffusion system. Existence and uniqueness of strong solution are proved for stochastic age-dependent population diffusion system in Hilbert space by use of Gronwall's lemma and Burkholder–Davis–Gundy's inequality.


Introduction
Stochastic differential equations has been more and more important in many fields, such as economics, biology, finance, ecology and other sciences (Caraballo & Liu, 1999;Chen, 2010Chen, , 2012DaPrato & Zabczyk, 2014;Liu, 2005). There has been many interests in application of age-dependent population systems in recent years. For example, Chou and Greenman (2016) and Assas, Dennis, Elaydi, Kwessi, and Livadiotis (2015) investigated hierarchical age-dependent populations with intra-specific competition and predation. Zhang (2008) considered the exponential stability of numerical solutions of the age-dependent population with diffusion in. Hernńdez-Cerón, Feng, and van den Driessche (2013) and Brauer (1999) provided thorough theoretical foundations for discrete and continuous-time age-dependent models. The optimal control and harvesting of the agestructured population system was investigated by Fister and Lenhart (2004). Pollard (1966), Block and Allen (2000) studied the effects of adding stochastic terms to discretetime age-dependent models that employed Leslie matrices. In Rathinasamy (2012), a class of split-step θ-method was proposed, and it is proved that the split-step θmethod converge to the analytical solutions of the equations under given conditions.
In this paper, we will focus on a class stochastic continuous time age-dependent model with diffusion. Actually, in Gurtnme (1973), the determined nonlinear agedependent population dynamic with diffusion can be CONTACT Qimin Zhang zhangqimin@nxu.edu.cn written in the following form.

∂P ∂t
where Q = (0, A) × (0, T), r ∈ [0, A] denotes the age, t ∈ [0, T] denotes the time, 0 < t < +∞, x ∈ denotes the position variables in space, ⊂ R N denotes a bounded region with smooth bounder ∂ ; P = P(r, t, x) ≥ 0 denotes the age-density function of population at time t age r and position x, Y(t, x) denotes the density function of population during [0, A] at time t and position x, k(r, t) ≥ 0 is the coefficient of population diffusion; μ 0 (r, t, x) ≥ 0 is the natural death rate function of population, μ e (r, t, x; p) ≥ 0 is the extra death rate function of population, for example, misfortune death etc.
denotes the Laplace operator with respect to the space variable. f (r, t, x; p) is the exterior disturbing function, as migration; β (r, t, x; p) is the birth rate of population; P 0 (r, x) ≥ 0 is the initial distribution of age-density of population at time t; A > 0 is the top age that an individual can survive in population.
Suppose that f (r, t, x; P) is stochastically perturbed, with f (r, t, x; P) → f (r, t, x; P) + g (r, t, x; P) where ω t = dω t /dt is the white noise. Then this environmentally perturbed system can be described by the stochastic partial differential equation as follows: Actually, Zhang and Han (2008) discussed this kind of age-dependent population diffusion system, where the mortality rate μ(r, t, x) is considered as a whole. In this paper, we will separate the mortality rate as two parts: the extra death rate and the natural death rate. Though it is a tiny change, but its biological meaning is very significant. Here we consider that only the extra death rate has a upper bound, which is a generalization of Zhang and Han (2008).
The rest of this paper is organized as follows. In Section 2, Many definitions and probationary knowledge are given to be ready for our main results. In Section 3, the existence and uniqueness of strong solution are proven by using of Gronwall's lemma and Burkholder-Davis-Gundy 's inequality. Conclusion will be proposed in Section 4.

Preliminaries
V is the dual space of V. We denote by · , | · | and · * the norms in V, H and V respectively; by ·, · the duality product between V, V , and by (·, ·) the scalar product in H, and there exists a constant m such that Let ω t be a Wiener process defined on complete probability space ( , F, P) and taking its values in the separable Hilbert space K, with increment covariance operator W. Let (F t ) t≥0 be the σ -algebras generated by {ω s , 0 ≤ s ≤ t}, then ω t is a martingale relative to (F t ) t≥0 and we have the following representation of ω t : where {e i } i≥0 is an orthonormal set of eigenvectors of W, β i (t) are mutually independent real Wiener processes with incremental covariance λ i > 0, We i = λ i e i and trW = ∞ i=1 λ i (tr denotes the trace of an operator (Pardoux, 1975)). For an operator B ∈ L(K, H) be the space of all bounded linear operators from K into H, we denote by B 2 the Hilbert-Schmidt norm, i.e., B 2 2 = tr(BWB T ). In this paper, ω t is a real standard Wiener process. Let Integrating on both sides of the first equation of system (2) on t, we can obtain the following nonlinear stochastic equation: where P t = P(r, t, x), P 0 = P(r, 0, x). So the study on the existence and uniqueness of strong solution of system (2) turns into the focus on system (3). In order to reach our expectations, some definitions and lemmas will be given first.
Definition 2.1: Let ( , F, {F t }, P) be a complete probability space with complete right continuous filtrations {F t } and ω t a Wiener process. Suppose that P 0 is a random variable such that E|P 0 | 2 < ∞. A stochastic process P t is said to be a strong solution on to the stochastic differential equation (3) for t ∈ [0, T] if the following conditions are satisfied: (1) P t is a F t -measurable random variable; ( Here C(0, T; H) denotes the space of all continuous functions from [0, T] to H; (3) Equation (3) is satisfied for every t ∈ [0, T] with probability one.
Let T be replaced by ∞, P t is called a global strong solution of (3). The objective in this paper is that, we hopefully find a unique process P t ∈ I p (0, T; V) ∩ L 2 ( ; C(0, T; H)) such that (3) hold. For this objective, we assume the following conditions are satisfied: (r, t, x) are continuous in Q, and there exists a positive constantsμ,β,k such that r, t, v) and g (r, t, v) are Lebesgue-measurable, ∀v ∈ L 2 H satisfying following condition (H): (H) There exist constants α > 0, ξ > 0, λ ∈ R, and a nonnegative continuous function γ (t), t ∈ R + , such that where, for arbitrary δ > 0, γ (t) satisfies γ (t) = o(e δt ), as t → ∞, i.e. lim t→∞ γ (t)/e δt = 0.
Remark 2.1: Observe that, owing to the continuity and subexponential growth of the term γ (t) e −ξ t , there exists a positive constantγ such that γ (t) e −ξ t ≤γ for all t ∈ R + , As a consequence, (H) implies a.e.t..

Existence of strong solutions
In order to prove the existence of solutions for Equation (3), we shall first prove the following lemmas.
We consider the equations μ e (r, s, x; P n s )P n s ds Lemma 3.1: {P n t } is a Cauchy sequence in L 2 ( ; C(0, T; H)).
Proof: For n > 1 and the process P n+1 t − P n t , it follows from Itô's formula where P n t := P n (rt, x), f (P n t ) := f (r, t, x; P n t ), g(P n t ) := g(r, t, x; P n t ). It is easy to deduce Consequently, inequality (8) yields Now, we estimate the terms on the right-hand side of inequality (8) by using the inequality 2ab ≤ a 2 l 2 + l 2 b 2 , l > 0.
On the other hand, we can get from Lipschisz condition In a similar manner, from Lipschisz condition we can obtain If we set then from inequalities (10)-(12), it could be deduced that there exists a positive constant c > 0 such that and consequently there exists k > 0 such that By iteration from inequality (14), we get Obviously, inequality (16) implies that {P n t } is a Cauchy sequence in L 2 ( ; C(0, T; H)).

Lemma 3.2:
The sequence {P n t } is bounded in I p (0, T; V).
Proof: Indeed, applying Itô's formula to |P n t | 2 with n ≥ 2 immediately yields Since {P n } is convergent in L 2 ( ; C(0, T; H)), it will be bounded in this space. Now, it is not difficult to check that there exists a positive constant k > 0 such that the righthand side of inequality (18) is bounded by this constant. We will estimate one of those terms. Firstly, we observe that Since {P n } is convergent in L 2 ( ; C(0, T; H)). Therefore, there exist a constant k such that T 0 E P n−1 s p ds ≤ k , and Lemma 3.2 is proved.
Theorem 3.1: Assume the preceding hypotheses hold, then there exists a process P t ∈ I p (0, T; V) ∩ L 2 ( ; C(0, T; H)) such that where f 1 ∈ I 2 (0, T; H), P 0 ∈ L 2 ( , F 0 , P; H) and M t is an H-valued continuous, square integrable F t -martingale. In addition, the following energy equality also holds: where M t denotes the quadratic variation of M t .
Proof: Use Lemmas 3.1 and 3.2, and use the same method in Métiver and Pellaumail (1980) we can get the conclusion.

Uniqueness of solutions
Now we will prove that there exists at most one solution of (3). This result will be deduced mainly from Itô's formula.

Conclusion
This paper has considered the existence and uniqueness of strong solution of a class of age-structured diffusion population system. By using of Burkholder-Davis-Gundy 's inequality and Itô's formula in Hilbert space and iteration of Cauchy sequence, the strong solution's existence and uniqueness of system (3) are obtained in I p (0, T; V) ∩L 2 ( ; C(0, T; H)).

Disclosure statement
No potential conflict of interest was reported by the authors.