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Existence of periodic solutions of a delayed predatory-prey system with functional response. (English) Zbl 1075.34067

The authors study the following periodic time-dependent predator-prey system with Holling type-III functional response \[ \frac{\displaystyle dN_1(t)}{\displaystyle dt}= N_1(t)[b_1(t)-a_1(t)N_1(t-\tau_1(t))-\frac{\displaystyle \alpha(t)N_1(t)}{\displaystyle 1+mN_1^2(t)}N_2(t-\sigma(t))],\tag{1} \]
\[ \frac{\displaystyle dN_2(t)}{\displaystyle dt}= N_2(t)[-b_2(t)+\frac{\displaystyle a_2(t)N_1^2(t-\tau_2(t))}{\displaystyle 1+mN_1^2(t-\tau_2(t))}], \] where \(N_1(t)\) and \(N_2(t)\) represent the densities of the prey and the predator at time \(t\), respectively. In system (1), \(b_i(t)\in C(\mathbb{R}, \mathbb{R})\), and \(a_i(t), \tau_i(t), \sigma(t), \alpha(t)\) are nonnegative, continuous, periodic functions with period \(T\), \(\int_{0}^{T}b_i(t)dt>0, \alpha(t)>0\), an \(m\) is a nonnegative constant. Using Gaines and Mawhin’s continuation theorem of coincidence degree theory [R. E. Gaines and J. L. Mawhin, Coincidence degree and nonlinear differential equations. Lecture Notes in Mathematics. 568. Berlin-Heidelberg-New York: Springer-Verlag (1977; Zbl 0339.47031)], sufficient conditions are derived for the existence of positive periodic solutions to system (1).

MSC:

34K13 Periodic solutions to functional-differential equations
92D25 Population dynamics (general)

Citations:

Zbl 0339.47031