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Hopf bifurcations in a predator-prey system with a discrete delay and a distributed delay. (English) Zbl 1206.34104
In this paper, the following predator-prey Lotka-Volterra system is studied
\[ x'(t) = x(t)[r_1-a_{11}x(t)-a_{12}y(t-\tau)], \]
\[ y'(t) = y(t)[-r_2 +a_{21}\int^t_{-\infty}G(t-s)x(s)ds - a_{22}y(t)], \] where the function \(G\) is bounded, non-negative, satisfying \(\int^{\infty}_0 G(s)ds = 1\) and \(\tau\), \(r_i\) and \(a_{ij}\) (\(i, j = 1, 2\)) are all positive constants. By introducing new variables \(u(t)\) and \(v(t)\) in terms of \(x(s)\) and \(y(s)\) over \(s\in (-\infty, t]\) respectively, the above system is transformed to a system of four equations about \((x(t), y(t), u(t), v(t))\) involving only one delayed term \(y(t-\tau)\). Then, by linearising the system at the positive equilibrium and analysing the associated characteristic equation, the asymptotic stability of the equilibrium is investigated and Hopf bifurcations are demonstrated. An explicit algorithm is given to determine the direction of Hopf bifurcations and the stability of bifurcating periodic solutions occurring through Hopf bifurcations. This is also verified by numerical simulations.

MSC:
34K60 Qualitative investigation and simulation of models involving functional-differential equations
34K18 Bifurcation theory of functional-differential equations
34K13 Periodic solutions to functional-differential equations
92D25 Population dynamics (general)
34K20 Stability theory of functional-differential equations
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References:
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