## Cooperative systems of differential equations with concave nonlinearities.(English)Zbl 0612.34035

The author considers the $$2\pi$$-periodic, cooperative system of differential equations $$x'=F(t,x)$$, assuming that $$D_ xF(t,x)$$ is continuous and $$(\partial F_ i/\partial x_ j)\geq 0$$ for $$i\neq j$$, that $$F_ i(t,x)\geq 0$$ for $$x\geq 0$$ and $$x_ i=0$$, that $$D_ xF(t,x)$$ is irreducible, and that $$D_ xF(t,x)\geq D_ xF(t,y)$$ for $$y>x>0$$. The principle theorem states: Every solution with $$x(t_ 0)\geq 0$$ can be continued to $$[t_ 0,\infty]$$ with x(t)$$\geq 0$$ for $$t\geq t_ 0$$. The statement of this theorem allows the possibility that x(t)$$\to \infty$$ as $$t\to \infty$$, though the author points out that it is easy to put conditions on F which rule out this possibility. In fact, it is sufficient that a single solution x(t) with $$x(t_ 0)>0$$, be bounded.
Important use is made of the Perron-Frobenious theory of nonnegative matrices, and of the results of Krasnoselkij concerning the uniqueness of the nonzero fixed point of a monotone operator which is strongly concave. The proof involves analysis of the Poincaré map $$Tx=\phi (2\pi;0,x)$$, where $$\phi$$ (t;s,x) is the solution which satisfies $$\phi (s;s,x)=x$$. It is first proved that $$DT(x)=(\partial \phi /\partial x)(2\pi;0,x)$$ has the monotonicity and concavity properties (M): $$DT(x)>0$$ if $$x>0$$, and (C): DT(y)$$\leq DT(x)$$ if $$0<x<y$$. the latter is proved by means of a theorem of Kamke. Properties (M) and (C) then imply: if $$T0=0$$, then for every $$x\geq 0$$, either $$T^ nx\to 0$$, or $$T^ nx\to \infty$$, or else T has a unique nonzero fixed point q such that $$q>0$$, and $$T^ nx\to q$$ as $$n\to \infty$$. Similar results apply if T0$$\neq 0$$. The paper contains a lengthy discussion of important applications of competitive and cooperative systems to mathematical models in ecology, epidemiology, economics, and biochemistry. There is also an interesting discussion of special models containing concave nonlinearities, and of previous work on these and similar problems.
Reviewer: A.L.Edelson

### MSC:

 34C25 Periodic solutions to ordinary differential equations 34C11 Growth and boundedness of solutions to ordinary differential equations 34D40 Ultimate boundedness (MSC2000)
Full Text:

### References:

 [1] Aronsson, G.; Kellogg, R.B., On a differential equation arising from compartmental analysis, Math. biosci., 38, 113-122, (1978) · Zbl 0375.34028 [2] Aronsson, G.; Mellander, I., A deterministic model in biomathematics, Math. biosci., 49, 207-222, (1980), Asymptotic behavior and threshold conditions · Zbl 0433.92025 [3] Coppel, W.A., Stability and asymptotic behavior of differential equations, (1965), Heath Boston · Zbl 0154.09301 [4] Hethcote, H.W.; Yorke, J.A., Gonorrhea transmission dynamics and control, () · Zbl 0542.92026 [5] Hirsch, M.W., Systems of differential equations which are competitive or cooperative. I: limit sets, SIAM J. math. analysis, 13, 167-179, (1982) · Zbl 0494.34017 [6] {\scHirsch} M.W., Systems of differential equations which are competitive or cooperative. II: Convergence almost everywhere, SIAM J. math. Analysis (to appear). [7] Hirsch, M.W., The dynamical systems approach to differential equations, Bull. am. math. soc., 11, 1-64, (1984) · Zbl 0541.34026 [8] Lajmanovich, A.; Yorke, J.A., A deterministic model for gonorrhea in a nonhomogeneous population, Math. biosci., 28, 221-236, (1976) · Zbl 0344.92016 [9] {\scKrasnoselskii} M.A., Translation along Trajectories of Differential Equations, Translations of Math. Monographs, Vol. 19, Am. Math. Soc., Providence, RI. [10] Krasnoselskii, M.A., Positive solutions of operator equations, (1964), Noordhoff Groningen [11] Selgrade, J.F., Asymptotic behavior of solutions to single loop positive feedback systems, J. diff. eqns, 38, 80-103, (1980) · Zbl 0419.34054 [12] {\scSmith} H.L., Periodic solutions of periodic competitive and cooperative systems, preprint. · Zbl 0609.34048 [13] Varga, R.S., Matrix iterative analysis, (1962), Prentice Hall Englewood Cliffs N.J · Zbl 0133.08602
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.