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**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 PoincarĂ© 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.

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 PoincarĂ© 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) |

### Keywords:

periodic first order differential; concave nonlinearities; competitive systems; Perron-Frobenious theory of nonnegative matrices; cooperative systems
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\textit{H. L. Smith}, Nonlinear Anal., Theory Methods Appl. 10, 1037--1052 (1986; Zbl 0612.34035)

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