GLOBAL STABILITY FOR DIFFERENTIAL EQUATIONS WITH HOMOGENEOUS NONLINEARITY AND APPLICATION TO POPULATION DYNAMICS P.Magal

In this paper we investigate global stability for a differential equation containing a positively homogeneous nonlinearity. We first consider perturbations of the infinitesimal generator of a strongly continuous semigroup which has a simple dominant eigenvalue. We prove that for ”small” perturbation by a positively homogeneous nonlinearity the qualitative properties of the linear semigroup persist. From this result, we deduce a global stability result when one adds a certain type of saturation term. We conclude the paper by an application to a phenotype structured population dynamic model.


1.
Introduction. The objective of this paper is to investigate the asymptotic behavior of solutions of abstract semilinear differential equations with homogeneous nonlinearities. The equation we consider has the form We suppose that A is the infinitesimal generator of a strongly continuous semigroup {T (t)} t≥0 of positive linear operators in a Banach lattice X, g is a nonlinear operator in X + satisfying g(cx) = cg(x), x ∈ X + , c ≥ 0, and F is a continuous linear functional satisfying F (x) > 0, ∀x ∈ X + \ {0} . Moreover, to assure the positivity of the solutions, we assume there exists ρ 0 (g) > 0 such that (g +ρ 0 (g) Id)(X + ) ⊂ X + .
Let us now consider the positively homogeneous Cauchy problem In section 2, we will recall a global center manifold theorem that will be used in section 3. In section 3, we will investigate the behavior of equation (2). More precisely, we assume that A has a dominating simple real eigenvalue λ 0 ∈ R associated to some positive eigenvectors φ 0 ∈ X + \ {0} and φ * 0 ∈ X * + \ {0} with φ * 0 (φ 0 ) = 1, with φ * 0 (φ) > 0, ∀φ ∈ X + \ {0} . Then (see Theorem 3.1) when the Lipschitz norm of g and ρ 0 (g) are small enough (A being fixed), there exists v g ∈ X + \ {0} with v g = 1 and there exists µ g ∈ R such that 542 P. MAGAL Moreover for each x ∈ X + \ {0} there exists α x > 0 such that v x (t) e µgt → α x v g as t → +∞. A usual way to prove such a result is to use the same technics as in Wysocki [15] and Takac [9] (see also references there in). In order to apply such a technic, some compactness arguments are necessary. Here we have not such a compactness property. So we use a direct approach related with the contraction of the semiflow due to the second eigenvalue of the linear semigroup. We refer to Webb [14] for a result going in this direction.
Since g is positively homogeneous one has the following relation between the solutions of equations (1) and (2) (see Proposition 3.3) In section 4, we will use equation (3) and results of section 3, to derive a global stability result for equation (1). The global stability result proved here is a general version of Theorem 4.11 p:236 in Magal and Webb [3]. An important argument to apply the technics used in [3] is the existence of a compact global attractor. In [3] the existence of global attractor is due to the compactness of the linear semigroup.
Here we avoid this problem and we allow some weaker compactness conditions on the linear semigroup. Finally, in section 5, we will apply the result to an example coming from population dynamics.

A reduction result.
In this section we state some results proved in Magal [5]. These results are variations of some of the results proved in Vanderbauwhede [10] [11], and in Vanderbauwhede and Iooss [12]. Let (X, . ) be a Banach space, let (A, D(A)) be the infinitesimal generator of a strongly continuous semigroup of bounded linear operators T (t), t ≥ 0, in X, and let g : X → X be a (nonlinear) operator from X into X, g is Lipschitz continuous. Then we want to reduce the following semi-linear problem: Assumption 2.1: a) X = X s ⊕ X c where X s and X c are closed subspaces, satisfying T (t)X s ⊂ X s and T (t)X c ⊂ X c , ∀t ≥ 0. We denote Π s ∈ L(X), and Π c ∈ L(X) the projection operators satisfying Π s (X) = X s , Π c (X) = X c , and Id − Π c = Π s . b) dim(X c ) < +∞, and X c ⊂ D(A) and σ (A c ) ⊂ Ri, where A c = AΠ c ∈ L(X). c) There exist β > 0 and M s ≥ 1 such that The center manifold will be related with function u ∈ C(R, X) solution of We denote Lip(X, X) = g : X → X : g Lip = sup x,y∈X:x =y and for all η ∈ R, The center manifold is defined for η ∈ (0, β), as follows: We refer to Magal [5] Theorem 1.3 for a proof of the following theorem.
Theorem 2.1. : Under Assumption 2.1. Let η ∈ (0, β) be fixed. Let δ 0 = δ 0 (A, η) > 0 be such that for each g ∈ Lip(X, X), with g Lip ≤ δ 0 , there exists a unique map Ψ : X c → X s which is Lipschitz continuous and such that We refer to Magal [5] Theorem 1.4 for a proof of the following theorem.
Theorem 2.2. : Under Assumption 2.1. Let η ∈ (0, β) be fixed. Then there exists 3. Homogeneous problem. Let X + be a positive cone of a Banach space X, i.e. X + closed convex subset of X such that i) λX + ⊂ X + , ∀λ ≥ 0, and ii) X + ∩−X + = {0} . Such a cone X + induces a partial order on X, denoted ≤ and defined by In the sequel, we denote X * the topological dual space of X (i.e. the space of continuous linear forms on X), and we denote X * + the dual cone defined by X * + = {ϕ ∈ X * : ϕ (x) ≥ 0, ∀x ∈ X + } . We recall that a bounded linear operator L ∈ L(X) is said to be positive if L(X + ) ⊂ X + .
Let (X, . ) be a Banach lattice with positive cone X + (see Schaefer [8]). In this section, we will make the following assumptions. Assumption 3.1: a) (A, D(A)) is the infinitesimal generator of a strongly continuous semigroup of positive bounded linear operators and there exist β > 0 and M ≥ 1, such that g is a nonlinear operator from X + to X, g is Lipschitz continuous on X + , and there exists ρ 0 (g) > 0 such that (g + ρ 0 (g) Id)(X + ) ⊂ X + . e) g is positively homogeneous, i.e. g(λx) = λg(x), ∀λ ≥ 0, ∀x ∈ X + . 544 P. MAGAL Remark: 1) Assumption 3.1 b) implies that the spectral bound of A satisfies s(A) = 0. 2) Assume that g is only Lipschitz continuous on bounded sets. Denote k > 0 the Lipschitz constant of g on B(0, 1) ∩ X + . Then if x, y ∈ B(0, M ) ∩ X + , by using Assumption 3.1 e) we have So, g is k−Lipschitz on X + . 3) Assumption 3.1 b) and c) will be verified if A is the infinitesimal generator of a strongly continuous semigroup of positive operator T (t), which is quasi-compact and irreducible i.e. ∀x ∈ X + \ {0} , and ∀ϕ ∈ X * + \ {0} , ∃t > 0 such that ϕ (T (t)x) > 0, and the spectral bound of A satisfies This result can be found in the book of Nagel [6] see Theorem 2.1 p:343, and remark d) p:344. But here we don't assume that φ 0 is a quasi-interior point. So the situation that we consider here is more general than the case of irreducible semigroups. We also refer to Webb [13] (see Proposition 2.3 and Proposition 2.5 and Remark 2.2) for a generalized version of this result, in which the compactness conditions on the linear semigroup are weaker.
Proof: i) − v) use classical arguments. We now prove vi). Let be x ∈ X + \ {0} . We have and since X is a Banach lattice we have and from Assumption 3.1 c) we have φ * 0 (x) > 0, and the result follows. The following theorem is the main result of this section.
We first note that and since g is positively homogeneous, we have Let be λ > 0. By Assumption 3.1 b), λ is in the resolvent set of A, and we denote R(λ, A) := (λI − A) −1 the resolvent operator of A. We have for all t, t 0 ∈ I, with t ≥ t 0 , are continuously differentiable. So from (22) we deduce that Furthermore, from (21) with t 0 = 0, and from (23), we deduce that the map t → T (t)y is continuously differentiable, so Finally by using (22), we have ∀t, t 0 ∈ I, with t ≥ t 0 , and since y ∈ D(A), we obtain α (t)y = α(t)Ay + α(t) g(y), ∀t ∈ I, and since y = 0, we deduce that there exists a certain constant µ ∈ R such that and µy = Ay + g(y).
Proof: This result is a direct consequence of Theorem 3.1.
We now prove some additional properties for the nonlinear eigenvalue µ g and the nonlinear eigenvector v g .
We denote .
Then one has P 0 ( v g ) = P 0 (φ 0 ), Moreover, since g is positively homogeneous, we also have but Aφ 0 = 0, so Consider now the equivalent norm which is well defined by Assumption 3.1 b). Then we have By using equations (40) (41)(43) and (44), we obtain ∀λ > 0, By using the fact that η < β, we deduce that there exist δ * 2 > 0 and 0 < C < 1 such that We have Furthermore, we have µ g v g = A v g + g( v g ) and since s(A) = 0, we have We conclude this section with a result relating solution of equations (1) and (2).  1 a), d), e), and assume that F ∈ X * + . Then for each x ∈ X + , the semilinear problem admits a unique mild solution u x ∈ C ([0, +∞) , X) which is given by Proof: We start by noting that the problem (46) has a unique v x (t). Also for λ > 0, we have so by applying Theorem 2.4 p:107 in Pazy [7], we deduce that We denote One can note that since x ∈ X + , and F ∈ X * + , u λ,x (t) and u x (t) are defined for all t ≥ 0. Moreover we have and since g is positively homogeneous, Let (X, . ) be a Banach lattice with positive cone X + . In this section, we will make the following assumptions.
and there exist β > 0 and M ≥ 1, such that We denote {T 0 (t)} t≥0 the strongly continuous semigroup of linear positive operators defined by T 0 (t) = e −s(A)t T (t), t ≥ 0, we denote {U τ (t)} t≥0 the strongly continuous semigroup of nonlinear operator from X + into X + solution of and we denote {S τ (t)} t≥0 the strongly continuous semigroup of nonlinear operator from X + into X + solution of , as t → +∞.
Assume in addition that s(A) > 0, and that g can be extended locally around each point of X + \ {0} by a continuously differentiable map. Then there exists τ ∈ (0, τ * ] such that for all τ ∈ [0, τ ] , we have s(A) + µ τ > 0, and u τ is exponentially asymptotically stable.
We now assume that s(A) > 0, and that g can be extended locally around each point of X + \ {0} by a continuously differentiable map. We now prove that for all τ > 0 small enough u τ is locally stable. From Proposition 3.2 we know that and since s(A) > 0, there exists τ * 1 > 0, such that

The generator of the linearized equation of equation (47) at u τ is given by
and by definition of u τ we have , ∀x ∈ X. Then Since g is differentiable on X + and X = X + − X + , we deduce that Indeed for h ∈ X + , x ∈ X + \ {0}, and ε > 0, we have From (50) we deduce that and from Proposition 3.2, we deduce that for each ε > 0, there exists τ ∈ (0, τ * 1 ] , such that B τ L(X) ≤ ε, ∀τ ∈ [0, τ ] .
It now remains to investigate the exponential asymptotic stability of the linear semigroup generated by , ∀x ∈ D(A).
But A 0 is the generator of T 1 (t) solution of where 0 < ε < min(β, s(A)).
(51) Then we have so, by using equations (51) and (52) we obtain and the exponential stability follows.

5.
Application to a population dynamics model. In this section we investigate a model which was already considered in Magal [4]. In this model we consider the evolution of a population with a continuous varying phenotype.
In (53) u = u(t, y) is the density of population with respect to a phenotype variable y ∈ (0, 1) at time t. The subpopulation of phenotype at time t in the range [y 1 , y 2 ] ⊆ (0, 1) is given by y2 y1 u(t, y)dy. The population is viewed as evolving over time due to the three separate processes of mutation, selection, and recombination. In (53) the mutation process is represented by the kernel operator γ [L(u) − u] , where γ is the mutation rate, and 0 ≤ α < 1 corresponds to a rate of movement in y per unit time for an individual which mutates. For example, if α = 1, the result of the mutation of individuals during a unit of time will give a constant distribution. In (53) the selection process for the population depends on the fitness of individuals with respect to the phenotype represented by the function β(y). Fitness is variable in y and the sign of β(y) may be positive or negative. In (53) there is also a density dependent mortality independent of phenotype represented by the crowing term 1 K 1 0 u(t)( y)d y. The problem (53) also incorporates DN A exchange in phenotype evolution represented by the term τ [R(u(t)) − u(t)] . The recombination operator R corresponds to the average rate at which two parent phenotypes y 1 and y 2 hybridize to yield offspring with phenotype y1+y2 2 . This form of recombination inheritance is an idealization and other genetic recombination processes could be treated in similar way. Problem (53) thus models the evolution of phenotype structure from the initial phenotype distribution u 0 ∈ L 1 + (0, 1) at time 0 subject to these processes.

P. MAGAL
We also refer to Magal and Webb [3] for similar model where the mutation process is represented by a diffusion operator with Neumann boundary conditions. We refer to the book by Burger [1] for a comprehensive and update treatment of this topics.
Theorem 5.1. R is a nonlinear operator from X + to X + satisfying the following properties: i) R is positive homogeneous, i.e. R(cφ) = cR(φ), ∀φ ∈ X + , ∀c ≥ 0; ii) R is Lipschitz continuous in X + ; We now recall some properties of the operator L. The following theorem is proved in [4] (see Theorem 2.2).
Theorem 5.2. The bounded linear operator L ∈ L (X, X) satisfies the following properties: i) L is compact; ii) L is irreducible; iii) The spectrum of L is σ (L) = α k : k = 0, 1, 2, ... ∪ {0} , and ∀k ≥ 0 the eigenvalue α k is simple; We are now interested in the linear part of the equation (53). So, we first consider the bounded linear operator The following theorem can be found in [4] (see Theorem 2.3).
We also recall that a strongly continuous semigroup T (t) on a Banach space X is called quasi-compact if lim t→+∞ dist(T (t), K(X)) = 0, where K(X) denotes the set of compact bounded linear operator on X, and for T ∈ L(X) dist(T, K(X)) = def inf { T − L : L ∈ K(X)} .
We are now in position to apply Theorem 2.1 p:343 , and remark (d) p:344 in Nagel [6], and vii) follows. We now return back to equation (53), and we apply Theorem 4.1.
Proof: The proof of this theorem is a direct consequence of Theorem 4.1, and Theorem 5.4.