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# Bautin bifurcation in delayed reaction-diffusion systems with application to the segel-jackson model

This research is supported by National Natural Science Foundation of China (11701120, 11771109) and the Foundation for Innovation at HIT(WH)

• In this paper, we present an algorithm for deriving the normal forms of Bautin bifurcations in reaction-diffusion systems with time delays and Neumann boundary conditions. On the center manifold near a Bautin bifurcation, the first and second Lyapunov coefficients are calculated explicitly, which completely determine the dynamical behavior near the bifurcation point. As an example, the Segel-Jackson predator-prey model is studied. Near the Bautin bifurcation we find the existence of fold bifurcation of periodic orbits, as well as subcritical and supercritical Hopf bifurcations. Both theoretical and numerical results indicate that solutions with small (large) initial conditions converge to stable periodic orbits (diverge to infinity).

Mathematics Subject Classification: Primary: 34K18; Secondary: 37L10.

 Citation: • • Figure 1.  The Bautin bifurcation diagram near the equilibrium for a) $l_2(0)<0$ and b) $l_2(0)>0$

Figure 2.  $a = 5$. a) the first Lyapunov coefficient $l_1$ varies with respect to $k$. b) the Bautin bifurcation diagram near the Bautin point $(k^\ast, \tau^\ast) = (0.3075, 0.6543)$

Figure 3.  $a = 5$. (a) Subcritical bifurcation diagram when $k = 0.8$. (b) Supercritical bifurcation diagram when $k = 0.1$

Figure 4.  $k = 0.1$ $a = 5$. For $\tau = 1.05$, the solution of (19) with initial conditions $u_0(x, t) = 0.3-0.16\cos x$, $v_0(x, t) = 0.5-0.16\cos x$ converges to a periodic solution

Figure 5.  $k = 0.1$, $a = 5$. For $\tau = 1.05$, the solution of (19) with initial conditions $u_0(x, t) = 10.3-0.16\cos x$, $v_0(x, t) = 10.5-0.16\cos x$ diverges to infinity

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