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# On oscillations to a 2D age-dependent predation equations characterizing Beddington-DeAngelis type schemes

• * Corresponding author: Peng Yang

The second author is supported by NSF grant of China (12071495, 11571382)

• In this study, a 2D age-dependent predation equations characterizing Beddington$-$DeAngelis type schemes are established to investigate the evolutionary dynamics of population, in which the predator is selected to be depicted with an age structure and its fertility function is assumed to be a step function. The dynamic behaviors of the equations are derived from the integrated semigroup method, the Hopf bifurcation theorem, the center manifold reduction and normal form theory of semilinear equations with non-dense domain. It turns out that the equations appear the oscillation phenomenon via Hopf bifurcation (positive equilibrium age distribution lose its stability and give rise to periodic solutions), as the bifurcation parameter moves across certain threshold values. Additionally, the explicit expressions are offered to determine the properties of Hopf bifurcation (the direction the Hopf bifurcation and the stability of the bifurcating periodic solutions). This technique can also be employed to other epidemic and ecological equations. Eventually, some numerical simulations and conclusions are executed to validating the major results of this work.

Mathematics Subject Classification: Primary: 34C20; Secondary: 37L10, 65L03, 92D25.

 Citation: • • Figure 1.  Numerical solutions of equations (BD) as $\tau = 5 < \tau_0 = 5.2509$: (a) image of the function $q(x)$; (b) solution structure of the predator; (c) solution structure of the prey; (d) phase portrait of equations (BD); (e) distribution function of the predator; (f) trajectory diagram of equations (BD)

Figure 2.  Numerical solutions of equations (BD) as $\tau = 7 > \tau_0 = 5.2509$: (a) image of the function $q(x)$; (b) periodic structure of the predator; (c) periodic structure of the prey; (d) phase portrait of equations (BD); (e) distribution function of the predator; (f) trajectory diagram of equations (BD)

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