Abstract
A number of empirical and theoretical studies shows that the exploitation of fish sources has benefitted a lot from artificial floating objects (abbr. FOBs) on the surface of ocean. In this paper we investigate the dynamical distribution in aggregations of tuna around two FOBs. We abandon the effort of precise computation for steady states and eigenvalues but utilize the monotonic intervals to determine the location of zeros and signs of eigenvalues qualitatively and use the symmetry of AS steady states to simplify the system. Our method enables us to find two more steady states than known results and complete the analysis of all steady states effectively. Furthermore, we display all bifurcations at steady states, including six bifurcations of co-dimension 1 and two bifurcations of co-dimension 2. One of bifurcations is a degenerate pitchfork bifurcation of co-dimension 4 but only a part of co-dimension 2 can be unfolded within the system. We construct sectorial regions to prove the nonexistence of closed orbits. Those results provide long-time prediction of steady numbers of tuna around the two FOBs and critical conditions for transitions of cases.
Highlights
It is a long period to use differential equations to model the spatio-temporal distribution and dynamics for wild animals
We show two pitchfork bifurcations, which occur at SS: (1, 1) near ν = N2(g) for 0 < g < 4 and g > 4 separately
In two sections we investigate bifurcations occurring at non-hyperbolic SS and non-hyperbolic AS
Summary
It is a long period to use differential equations to model the spatio-temporal distribution and dynamics for wild animals (see, e.g., [1, 7, 13]). We discuss (22) in the following: (1.3i): If 0 < g < 16 and ν > N2(g), by (S3) of Lemma 2.2 and (23) we get λ1 < 0 < λ2, implying that the SS is a saddle, as indicated on lines 3 and 11 of Table 1.
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