Competitive and Cooperative Systems: A Mini-review

The theory of competitive and cooperative dynamical systems has had some remarkable applications to the biological sciences. The interested reader may consult the monograph [28] and lecture notes [29] of Smith, and to a forthcoming review by the authors [11] for a more in-depth treatment. 1 Strong Monotonicity for ODEs In this brief review we give some of the main results in the theory of competitive and cooperative systems. But first, we give some new strong monotonicity results for odes. Let J be a nontrivial open interval, D ⊂ IR be an open set, f : J × D → IR be a locally Lipschitz function, and consider the ordinary differential equation x′ = f(t, x) (1) Denote by x(t, t0, x0) the non-continuable solution of the initial value problem x(t0) = x0 for t0 ∈ J . A cone K in IR is a non-empty, closed subset of IR satisfying K+K ⊂ K, IR · K ⊂ K and K ∩ (−K) = {0}. We hereafter assume K nonempty interior in IR. The order relations ≤, <,¿ are induced by K as follows: x ≤ y if and only if y−x ∈ K; x < y if x ≤ y and x 6= y, and x ¿ y whenever y−x ∈ IntK. A cone is a polyhedral cone if it is the intersection of a finite family of half spaces. The standard cone IR+ = ∩i=1{x : 〈ei, x〉 ≥ 0} is polyhedral (ei is the unit vector in the xi-direction) while the ice cream cone K = {x ∈ IR : x1 +x 2 2 + · · ·+xn−1 ≤ xn, xn ≥ 0} is not. The dual cone K∗ is the set positive linear functionals, i.e., linear functionals λ ∈ (IRn)∗, the dual space of IR, such that λ(K) ≥ 0. If we adopt the standard inner product 〈, 〉 on IR then we can identify (IRn)∗ with IR since for each λ ∈ K∗ we can find a ∈ IR such that λ(x) = 〈a, x〉 for all x. We use the following easy result; see e.g. Walcher [35]. 2 Morris W. Hirsch and Hal L. Smith Lemma 1. Let x ∈ K. Then x ∈ IntK if and only if λ(x) > 0 for all λ ∈ K∗ \ {0}. We say that (1) is monotone, or order-preserving, if whenever x0, x1 ∈ D satisfy x0 ≤ x1 and the solutions x(t, t0, x0) and x(t, t0, x1) are defined on [t0, t1], t1 > t0, then x(t, t0, x0) ≤ x(t, t0, x1) holds for t ∈ [t0, t1]. The vector field f : J × D → IR is said to satisfy the quasimonotone condition in D if for every (t, x), (t, y) ∈ J ×D we have (Q)x ≤ y and φ(x) = φ(y) for some φ ∈ K∗ implies φ(f(t, x)) ≤ φ(f(t, y)). The quasimonotone condition was introduced by Schneider and Vidyasagar [25] for finite dimensional, autonomous linear systems and used later by Volkmann [34] for nonlinear infinite dimensional systems. The following result is certainly inspired by a result of Volkmann [34] and work of W. Walter [?]. See also Uhl [33] and Walcher [35]. The proof appears in [11]. Theorem 1.1 Let f satisfies (Q) in D, t0 ∈ J , and x0, x1 ∈ D. Let t0 are such that both x(t, t0, xi), i = 0, 1 are defined, then x(t, t0, x0) t0. A matrix A is K-nonnegative if A(K) ⊂ K. Corollary 1.1 says that X(t) is K-nonnegative for t ≥ t0 if (M) holds. The domain D is p-convex if for every x, y ∈ D satisfying x ≤ y the line segment joining them also belongs to D. Let ∂f ∂x (t, x) be continuous on J ×D. We say that f (or (1)) is K-cooperative if for all t ∈ J, y ∈ D, (M) holds for the function x → ∂f ∂x (t, y)x. By Corollary 1.1 applied to the variational equation X ′(t) = ∂f ∂x (t, x(t, t0, x0))X, X(t0) = I we conclude that if f is K-cooperative then X(t) = ∂x ∂x0 (t, t0, x0) is K-positive. Straightforward arguments lead to the following result. Competitive and Cooperative Systems: a mini-review 3 Theorem 1.2 Let ∂f ∂x (t, x) be continuous on J ×D. Then (Q) implies that f is K-cooperative. Conversely, if D is p-convex and f is K-cooperative, then (Q) holds. If K = IR+, then it is easy to see by using the standard inner product that we may identify K∗ with K. The quasimonotone hypothesis reduces to the Kamke condition [22, 14]: x ≤ y and xi = yi implies that fi(t, x) ≤ fi(t, y). This holds by taking φ(x) = 〈ei, x〉 and noting that every φ ∈ K∗ can be represented as a positive linear combination of these functionals. If f is differentiable, the Kamke condition implies ∂fi ∂xj (t, x) ≥ 0, i 6= j. (3) Conversely, if ∂f ∂x (t, x) is continuous on J × D and satisfies (3) and if D is p-convex, then the Kamke condition holds. Stern and Wolkowicz [32] give necessary and sufficient conditions for (M) to hold for matrix A relative to the ice cream cone K = {x ∈ IR : x1 + x2 + · · · + xn−1 ≤ xn, xn ≥ 0}. Let Q denote the n × n diagonal matrix with first n − 1 entries 1 and last entry −1. Then (M) holds for A if and only if QA + A Q + αQ is negative semidefinite for some α ∈ IR. Their characterization extends to other ellipsoidal cones. Additional hypotheses are required for establishing the strong order preserving property and here we provide full details following [11]. Recall that the matrix A is strongly positive if A(K \ {0}) ⊂ IntK. The following hypothesis for the matrix A follows Schneider and Vidyasagar [25]. (T)for all x 6= 0, x ∈ ∂K, there exists ν ∈ K∗ such that ν(x) = 0 and ν(Ax) > 0. Our next result was proved by Elsner [3] for the case of constant matrices, answering a question in [25]. Our proof follows that of Theorem 4.3.26 of Berman et al [1]. Proposition 1.1 Let the linear system (2) satisfy (M). Then the fundamental matrix X(t1) is strongly positive for t1 > t0 if there exists s satisfying t0 < s ≤ t1 such that (T) holds for A(s). Proof: If not, there exists x > 0 such that the solution of (2) given by y(t) = X(t)x satisfies y(t1) ∈ ∂K \ {0}. By Corollary 1.1, y(t) > 0 for t ≥ t0 and y(t) ∈ ∂K for t0 ≤ t ≤ t1. Let s ∈ (t0, t1] be such that (T) holds for A(s). Then there exists ν ∈ K∗ such that ν(y(s)) = 0 and ν(A(s)y(s)) > 0. As ν ∈ K∗ and y(t) ∈ K, h(t) := ν(y(t)) ≥ 0 for t0 ≤ t ≤ t1. But h(s) = 0 and d dt |t=sh(t) = ν(A(s)y(s)) > 0 which, taken together, imply that h(s− δ) < 0 for small positive δ, giving the desired contradiction. Proposition 1.2 leads immediately to a result on strong monotonicity for the nonlinear system (1). 4 Morris W. Hirsch and Hal L. Smith Theorem 1.3 Let D be p-convex, ∂f ∂x (t, x) be continuous on J × D, and f be K-cooperative. Let B = {(t, x) ∈ J ×D : (T) does not hold for ∂f ∂x (t, x)}. Suppose that for all (t0, x0) ∈ J × D, the set {t > t0 : (t, x(t, t0, x0)) ∈ B} is nowhere dense. Then x(t, t0, x0) ¿ x(t, t0, x1) for t > t0 for which both solutions are defined provided x0, x1 ∈ D satisfy x0 < x1. In particular, this holds if B is empty. Proof: We apply the formula x(t, t0, x1)− x(t, t0, x0) = ∫ 1