Exponential Ordering Induced Monotonicity

2.6 Exponential Ordering Induced Monotonicity

Reaction–diffusion equations with delayed reaction terms and, more generally, abstract functional differential equations have been widely used to model the evolution of a physical system distributed over a spatial domain [408]. In the celebrated work of Martin and Smith [243,244], the monotonicity of the semi-flow generated by an abstract functional differential equation was established, and the powerful theory of monotone dynamical systems was applied to obtain some detailed descriptions of the generic dynamics of the semiflow. In order for the semiflow to be order-preserving with respect to the pointwise ordering of the phase space, the aforementioned work requires that the nonlinear reac-tion term satisfy a certain quasi-monotonicity condireac-tion, which, in the special case of a reaction–diffusion equation with delay, requires that the reaction term be monotone and thus limits the applications in some cases. It is there-fore natural to ask whether the quasi-monotonicity condition in the work of Martin and Smith can be relaxed. This question was addressed in Smith and Thieme [329, 331] for the case of ordinary functional differential equations (that is, the spatial diffusion is absent), where they established the mono-tonicity of the semiflow in a restricted but sufficiently large subspace with a nonstandard exponential ordering. In this section we extend the exponen-tial ordering and its induced monotonicity to abstract functional differenexponen-tial equations and delayed reaction–diffusion equations. These results will be used to obtain threshold dynamics for a nonlocal and delayed reaction–diffusion population model in Chapter9.

Let A : Dom(A)→ X be the infinitesimal generator of an analytic semi-group T (t) satisfying T (t)P ⊂ P, ∀t ≥ 0. For convenience, we denote T (t) by e^At. Let r≥ 0 be fixed and let C := C([−r, 0], X). For μ ≥ 0, we define

K_μ={φ ∈ C : φ(s) ≥X 0,∀s ∈ [−r, 0], and φ(t) ≥X e(A−μI)(t−s)φ(s),

∀0 ≥ t ≥ s ≥ −r}.

Then Kμ is a closed cone in C. Let≥μ be the partial ordering on C induced by Kμ. The meaning of≤μ and≤X should be clear.

Lemma 2.6.1. Assume that φ ∈ C is differentiable on (−r, 0) and φ(t) ∈ Dom(A),∀t ∈ (−r, 0). Then φ ≥μ0 if and only if

φ(−r) ≥X0, and dφ(t)

dt − (A − μI)φ(t) ≥X 0,∀t ∈ (−r, 0).

Proof. Assume that φ≥μ0; that is, φ∈ Kμ. It then follows that φ(−r) ≥X0, and for any t∈ (−r, 0) and h > 0 with t + h ∈ [−r, 0],

φ(t + h)− φ(t)

h ≥X

e^(A−μI)hφ(t)− φ(t)

h .

Since φ is differentiable at t and φ(t)∈ Dom(A), letting h → 0⁺ and using the definition of infinitesimal generators (see, e.g., [272]), we get

70 2 Monotone Dynamics dφ(t)

dt = lim

h→0⁺

φ(t + h)− φ(t)

h ≥X(A− μI)φ(t), ∀t ∈ (−r, 0).

Conversely, assume that φ(−r) ≥X 0 and ^dφ_dt^(t)−(A−μI)φ(t) ≥X 0,∀t ∈ (−r, 0). Let t ∈ (−r, 0] be fixed. Clearly, the function u(s) := e(A−μI)(t−s)φ(s) is differentiable for s∈ (−r, t). By the property of analytic semigroups (see, e.g., [150,272]) and the positivity of e(A−μI)(t−s)= e^−μ(t−s)e^A(t−s), we have

du(s)

ds =−(A − μI)e(A−μI)(t−s)φ(s) + e(A−μI)(t−s)dφ(s) ds

=−e(A−μI)(t−s)(A− μI)φ(s) + e(A−μI)(t−s)dφ(s) ds

= e(A−μI)(t−s) dφ(s)

ds − (A − μI)φ(s)



≥X 0.

Thus we get φ(t)− e(A−μI)(t−s)φ(s) = u(t)− u(s) =t s

du(τ)

dτ dτ ≥X 0, ∀s ∈ [−r, t]. This, together with φ(−r) ≥X0, implies φ∈ Kμ.

Let σ > 0 and let u : [−r, σ) → X be a continuous map. For each t ∈ [0, σ), we define ut∈ C by ut(s) = u(t + s),∀s ∈ [−r, 0]. Let D be an open subset of C. Assume that F : D→ X is continuous and satisfies a Lipschitz condition on each compact subset of D. We consider the abstract functional differential equation

du(t)

dt =Au(t) + F (ut), t > 0, u₀=φ∈ D.

(2.1)

By the standard theory (see, e.g., [243,408]), for each φ∈ D, equation (2.1) admits a unique mild solution u(t, φ) on its maximal interval [0, σφ). Moreover, if σφ> r, then u(t, φ) is a classical solution to (2.1) for t∈ (r, σφ). In order to get a monotone solution semiflow of (2.1) with respect to≥μ, we will impose the following monotonicity condition on F :

(M_μ) μ(ψ(0)− φ(0)) + F (ψ) − F (φ) ≥X 0 for φ, ψ∈ D with φ ≤μψ.

Theorem 2.6.1. Let (M_μ) hold. If φ≤μψ, then u_t(φ)≤μu_t(ψ) for all t≥ 0 such that both solutions are defined.

Proof. Let v^∗∈ int(P ) be fixed. For any  > 0, define F(φ) := F (φ)+ v^∗for φ∈ D, and let u^(t, ψ) be the unique mild solution of the following equation

du(t)

dt =Au(t) + F(ut), t > 0, u₀=ψ∈ D.

(2.2)

Without loss of generality, we assume that u(t, φ) and u^(t, ψ) are both de-fined on [0,∞) (if not, we replace [0, ∞) by the intersection of their maximal intervals of existence). Let y^(t) := u^(t, ψ)− u(t, φ) and define

2.6 Exponential Ordering Induced Monotonicity 71

P ={t ∈ [0, ∞) : y^t≥μ 0}.

Clearly, P is closed and 0 ∈ P . We claim that if t0 ∈ P , then there exists δ₀ > 0 such that [t₀, t₀+ δ₀] ⊂ P . Indeed, by the abstract integral forms of equations (2.1) and (2.2), we have

y^(t) = e(A−μI)(t−s)y^(s) +

 t s

e(A−μI)(t−τ) (2.3)

(F (u^_τ(ψ))− F (uτ(φ)) + μ (u^(τ, ψ)− u(τ, φ)) + v^∗) dτ for all t≥ s ≥ 0. By the condition u^t₀(ψ)≥μut₀(φ) and assumption (Mμ), it then follows that

(F (u^_t(ψ))− F (ut(φ)) + μ (u^(t, ψ)− u(t, φ)) + v^∗)|t=t0 ≥Xv^∗X0.

Thus there exists δ₀> 0 such that

F (u^_t(ψ))− F (ut(φ)) + μ (u^(t, ψ)− u(t, φ)) + v^∗≥X0,∀t ∈ [t0, t₀+ δ₀].

By the integral equation (2.3) and the positivity of the semigroup e^(A−μI)t, we then get

y^(t)≥Xe(A−μI)(t−s)y^(s), ∀t₀≤ s ≤ t ≤ t₀+ δ₀,

which, together with the definition of (Kμ), implies that u^_t(ψ)≥μut(φ), ∀t ∈ [t₀, t₀+ δ₀].

Let P₁:={t : [0, t] ⊂ P }. We claim that sup P1=∞. Assume, by way of contradiction, that t^∗= sup P₁<∞. Then there is a sequence {tn} ⊂ P1⊂ P such that tn → t^∗. Thus the closedness of P implies that t^∗ ∈ P . By the claim in the previous paragraph, [t^∗, t^∗ + δ^∗] ⊂ P for some δ^∗ > 0, and hence t^∗+ δ^∗ ∈ P1, which contradicts the definition of t^∗. It then follows that [0,∞) ⊂ P , and hence P = [0, ∞).

By a standard argument, we have lim_→0+u^_t(ψ) = ut(ψ),∀t ≥ 0. Letting

 → 0⁺ in y_t^ = u^_t(ψ)− ut(φ) ≥μ 0, we get ut(ψ)− ut(φ) ≥μ 0, and hence ut(ψ)≥μut(φ),∀t ≥ 0.

For simplicity, in the rest of this section we assume that for each φ∈ C, equation (2.1) admits a unique mild solution u(t, φ) defined on [0,∞). Then (2.1) generates a semiflow on C by Φ(t)(φ) = ut(φ), φ∈ C. Clearly, condition (Mμ) is sufficient for Φ(t) : C → C to be monotone with respect to ≤μ

in the sense that Φ(t)(φ) ≤μ Φ(t)(ψ) whenever φ ≤μ ψ and t ≥ 0. In some applications of monotone dynamical systems, however, we need a strong order-preserving property (see, e.g., [326]). The semiflow Φ(t) : C → C is said to be strongly order-preserving with respect to ≤μ if it is monotone and if whenever φ <_μψ, there exist open subsets U, V of C with φ∈ U and ψ ∈ V and t₀> 0 such that Φ(t₀)(U )≤μΦ(t₀)(V ). Next we show that the following slightly stronger condition than (M_μ) is sufficient for Φ(t) to be strongly order-preserving:

72 2 Monotone Dynamics

(SM_μ) μ(ψ(0)− φ(0)) + F (ψ) − F (φ) X 0 for φ, ψ∈ C with φ ≤μψ and φ(s)X ψ(s),∀s ∈ [−r, 0].

Theorem 2.6.2. Assume that T (t)(P \ {0}) ⊂ int(P ), ∀t > 0, and (SMμ) holds. Then the solution semiflow Φ(t) is strongly order-preserving on C with respect to≤μ.

Proof. Let v^∗∈ int(P ) be fixed, and define φ^∗∈ C by φ^∗(t) = e(A−μI)(t+r)v^∗,∀t ∈ [−r, 0].

Then φ^∗(s) X 0,∀s ∈ [−r, 0], and Lemma 2.6.1 implies that φ^∗ ≥μ 0.

For any ψ ∈ C, the sequence of points ψn = ψ + _n¹φ^∗ in C satisfies ψ <_μ ψ_n+1 <_μ ψ_n,∀n ≥ 1, and ψn → ψ as n → ∞. By this property and the continuity of F , it is easy to see that (SM_μ) implies (M_μ). Then we conclude from Theorem2.6.1 that Φ(t) is monotone on C. Moreover, for each φ∈ C, u(t, φ)∈ Dom(A), ∀t > r. For every φ <μ ψ, the strong positivity of T (t) = e^Atimplies that φ(0) <Xψ(0), and hence, in view of ut(φ)≤μut(φ), ∀t ≥ 0, we have u(t, φ) X u(t, ψ) for all t > 0. Fix a real number t₀ > 2r and let φ⁰ <μ ψ⁰ be given. By condition (SMμ), the continuity of F , and the compactness of [t₀− r, t0], it then follows that there is a sufficiently small

₀> 0 such that

F (ut(ψ⁰))− F (ut(φ⁰)) + μ

u(t, ψ⁰)− u(t, φ⁰)

≥X ₀v^∗, ∀t ∈ [t0− r, t0].

Since

(φ,ψ)→(φlim⁰,ψ⁰)(u(t₀− r, ψ) − u(t0− r, φ)) = u(t0− r, ψ⁰)− u(t0− r, φ⁰)X 0

and

(φ,ψ)→(φlim⁰,ψ⁰)F (ut(ψ))− F (ut(φ)) + μ (u(t, ψ)− u(t, φ))

= F (ut(ψ⁰))− F (ut(φ⁰) + μ

u(t, ψ⁰)− u(t, φ⁰) uniformly for t∈ [t₀− r, t₀], there exist open subsets U, V of C with φ⁰ ∈ U and ψ⁰ ∈ V such that for every φ ∈ U and ψ ∈ V , we have u(t0− r, ψ) − u(t₀− r, φ) X 0 and

d(u(t, ψ)− u(t, φ))

dt − (A − μI)(u(t, ψ) − u(t, φ))

= F (ut(ψ))− F (ut(φ)) + μ (u(t, ψ)− u(t, φ)) X 0, ∀t ∈ [t0− r, t0].

Note that u(t, φ) and u(t, ψ) are both classical solutions for t > r. By Lemma 2.6.1, we then get u_t0(ψ)− ut₀(φ) ≥μ 0,∀ψ ∈ V, φ ∈ U, and hence u_t0(U )≤μu_t0(V ).

2.7 Notes 73 Note that in the case where X =R and A is the zero operator, ≤μreduces to the exponential ordering introduced by Smith and Thieme [329] for scalar non-quasi-monotone ordinary delay differential equations.

Let (Xi, Pi), 1 ≤ i ≤ n, be ordered Banach spaces with int(Pi) = ∅, and let Ai : Dom(Ai) → Xi be the infinitesimal generator of an analytic semigroup Ti(t) satisfying Ti(t)Pi ⊂ Pi,∀t ≥ 0. Let X = "n

i=1Xi, P =

"n

i=1Pi, T (t) = "n

i=1Ti(t), A = "n

i=1Ai, Dom(A) = "n

i=1Dom(Ai). Then A : Dom(A)→ X is the infinitesimal generator of the analytic semigroup T (t) defined on the ordered Banach space (X, P ). Let B = (bij) be an n×n matrix with bij ≥ 0, ∀1 ≤ i = j ≤ n. Define

KB={φ ∈ C : φ(s) ≥X0,∀s ∈ [−r, 0], and φ(t) ≥Xe^A(t−s)e^B(t−s)φ(s),

∀ 0 ≥ t ≥ s ≥ −r}.

Then K_B is a closed cone in C and induces a partial order≥B on C.

Remark 2.6.1. By an argument similar to that in Theorem2.6.1, we can prove that the solution semiflow of (2.1) is monotone with respect to≤B under the following monotonicity condition:

(MB) F (ψ)− F (φ) ≥XB(ψ(0)− φ(0)) for φ, ψ∈ D with φ ≤Bψ.

Clearly, in the case where n = 1 and B =−μ, ≥B reduces to≥μ. Replac-ing −μ with B in (SMμ), we get a stronger condition (SM_B). By a similar argument as in Theorem2.6.2, we should be able to prove that the solution semiflow of (2.1) is strongly order-preserving with respect to≤Bunder (SM_B) and an additional irreducibility assumption. For the details in the special case where X =Rⁿ and A = 0, we refer to [331].

2.7 Notes

There have been extensive investigations on monotone dynamical systems (see, e.g., Hess [152], Smith [326] and the references therein). For strongly monotone continuous-time dynamical systems one has generic convergence:

There is an open and dense subset of the phase space such that any orbit emanating from it converges to an equilibrium (see Hirsch [160], Pol´a˘cik [279]

and Smith and Thieme [330]). However, for strongly monotone discrete-time dynamical systems there is no generic convergence to fixed points; see, e.g., Tak´a˘c [353] and Dancer and Hess [87] for counterexamples in periodic differen-tial equations, the Poincar´e (period) maps of which define strongly monotone discrete-time dynamical systems. It is well known that for smooth strongly monotone discrete-time dynamical systems one has generic convergence to cycles (see Pol´a˘cik and Tere˘s˘c´ak [280,281]).

Theorem 2.1.1is due to Dancer [84]. Remark 2.1.1seems to be new. Re-mark2.1.3is due to Hsu, Smith and Waltman [174]. Theorem2.1.2is due to Zhao and Jing [444], which is a generalization of Smith [318, Theorem 2.1].

74 2 Monotone Dynamics

Theorem 2.2.1 extends Smith [326, Theorem 2.3.1] on strongly order-preserving continuous-time semiflows to monotone semiflows. A related result is Jiang and Yu [193, Theorem 3] on global asymptotic order stability for monotone maps on a strongly ordered space X with the property that every nonempty and compact subset has both a greatest lower bound and a least upper bound in X. Theorem2.2.2is due to Zhao [432]. Theorem 2.2.3seems to be new and is a variant of Smith [326, Theorem 2.3.2]. In the proof of Theo-rem2.2.3, we have used Theorem1.2.2for global convergence. Theorem2.2.4 seems to be new and extends Ogiwara and Matano [265, Theorem 2.4] on local convergence to global convergence. Tak´a˘c [355] also investigated conver-gence to a fixed point for a class of strongly monotone discrete-time dynamical systems in a strongly ordered Banach space.

Theorems 2.3.1, 2.3.2, and2.3.3 are due to Hirsch [162]. Condition (C3) was introduced by Zhao [432, Lemma 1] for uniqueness of positive fixed points.

Lemma 2.3.2 is due to Zhao [432], and Theorem 2.3.4is a generalization of [432, Theorem 2.3]. Tak´a˘c [349] established global convergence for subhomoge-neous (sublinear) and strongly monotone maps, which is an extension of a re-sult in Smith [317] concerning monotone and concave maps. Jiang [191] proved convergence for finite-dimensional monotone and subhomogeneous (sublinear) discrete-time dynamical systems. This result was generalized by Wang [383]

to the Poincar´e maps associated with periodic subhomogeneous and quasi-monotone reaction–diffusion systems subject to Neumann boundary condi-tions, and by Wang and Zhao [387] to monotone and subhomogeneous dis-crete dynamical systems on product Banach spaces. Monotone and strictly subhomogeneous (sublinear) semiflows generated by cooperative systems of functional differential equations and quasi-monotone reaction–diffusion sys-tems with delays were studied by Zhao and Jing [444] and Freedman and Zhao [124], respectively. Theorems2.2.2and2.3.2were applied to a nonlocal reaction–diffusion model by Freedman and Zhao [125]. The part metric was introduced by Thompson [373]. Krause and Nussbaum [205] proved a limit set trichotomy for part metric contractive maps on solid and normal cones in Banach spaces, and made a very interesting observation that a monotone map with strong subhomogeneity is contractive for the part metric on the interior of the cone. Tak´a˘c [354] also utilized the concept of part metric for convergence in discrete dynamical systems. Theorem2.3.5and Remarks2.3.2 and 2.3.3 are due to Zhao [438]. For global convergence in monotone and uniformly stable skew-product semiflows, we refer to Jiang and Zhao [194].

Theorem2.4.1and Proposition2.4.1are due to Hsu, Smith and Waltman [174]. In the proof of Theorem 2.4.1, we have used Theorem1.2.2for global convergence. The notion of compression was introduced by Hess and Lazer [154]. Theorem2.4.2is a generalization of a result in [154]. In the proof of The-orem2.4.2, again we have used Theorem1.2.2. Smith and Thieme [332] studied stable coexistence and bistability for competitive continuous-time semiflows on ordered Banach spaces, and showed that a “thin” separatrix separates the basins of attraction of the two locally stable single-population steady states

2.7 Notes 75 under the assumption that the coexistence steady state is unique. Wang and Jiang [384, 385] also obtained some general properties for strongly competi-tive discrete-time dynamical systems on strongly ordered topological vector spaces.

Section 2.5 is adapted from Jiang, Liang and Zhao [195], where these results were also applied to three reaction–diffusion systems modelling man-environment-man epidemics, single-loop positive feedback, and two-species competition, respectively. The concepts of upper and lower boundaries, order decomposition, and d-hypersurface were introduced by Hirsch [159] and well developed by Tak´a˘c [350, 351], Wang and Jiang [385], and Liang and Jiang [224]. Tak´a˘c [355] also employed the d-hypersurface to study the convergence for monotone discrete-time dynamical systems and two-species periodic com-petitive reaction–diffusion systems.

Section2.6is taken from Wu and Zhao [411] and was motivated by Smith and Thieme [329,331], where a nonstandard positive cone was introduced and applied to non-quasi-monotone ordinary differential equations and systems with delays. The exponential ordering was also used earlier by Hadeler and Tomiuk [140] to show the existence of nontrivial periodic solutions of a class of scalar difference–differential equations.

The theory of abstract competitive systems has found nontrivial applica-tions to two-species Lotka–Volterra competition reaction–diffusion systems, see, e.g., He and Ni [147], Lou, Xiao and Zhou [235], Zhao and Zhou [447]

and the references therein. Hsu and Zhao [172] also gave a complete classi-fication for the global dynamics of a two-species Lotka–Volterra competition model with seasonal succession. For abstract competitive systems, Lam and Munther [210] obtained two sufficient conditions that guarantee, in the ab-sence of coexistence steady states, the global asymptotic stability of one of two semitrivial steady states.

The monotone dynamical systems approach to traveling waves and spread-ing speeds has been well developed for discrete- and continuous-time evolu-tion systems admitting the comparison principle, we refer to Weinberger [401], Lui [239], Weinberger [402], Li, Weinberger and Lewis [220], Liang and Zhao [225, 226], Liang, Yi and Zhao [227], Fang and Zhao [110], Ding and Liang [97] for the theory of monostable waves and spreading speeds; Fang and Zhao [111] for the general theory of bistable waves.

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