Distribution of SS and AS and their bifurcations in aggregations of tuna around two FOBs

Shaowen Shi; Weinian Zhang

doi:10.3934/dcdsb.2019043

Article Contents

2019, Volume 24, Issue 9: 5041-5081. Doi: 10.3934/dcdsb.2019043

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Distribution of SS and AS and their bifurcations in aggregations of tuna around two FOBs

Shaowen Shi and
Weinian Zhang^,

Yangtze Center of Mathematics and Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

^*Corresponding author: Weinian Zhang
^*Corresponding author: Weinian Zhang

Received: July 2018

Revised: October 2018

Early access: February 2019

Published: July 2019

Supported by NSFC grants #11771307, #11726623 and #11521061 and PCSIRT IRT-15R53.

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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.

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Mathematics Subject Classification: Primary: 34C23; Secondary: 34C05, 37G10, 92B05.

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Figure 1. Parameter regions for qualitative properties. The left comes from [12] and the right from Table 2

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Figure 2. Graph of $P_3$ with $g = 20$ and $\nu = 15$

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Figure 3. Cusp bifurcation at the SS: $ (\sqrt3, \sqrt3) $

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Figure 4. Degenerate pitchfork bifurcation at the SS: $ (1, 1) $

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Figure 5. Bifurcation diagriam of system (67)

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Figure 6. Phase plane of system (7) in (C3)

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Figure 7. Phase plane of system (7) in (C4)

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Figure 8. Phase plane of system (7) in (C5).

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Figure 9. Parameter plane of system (7)

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Figure 10. Phase diagrams of system (7)

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Figure 11. Phase diagrams of system (7)

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Figure 12. Phase diagrams of system (7)

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Figure 13. Phase portraits of system (7)

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Figure 14. Phase portraits of system (7)

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Figure 15. Phase portraits of system (7)

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Table 1. Number and stability of steady states found in [12]

$ g $	$ \nu $	number and stability
$ g<4 $	$ \nu<N_2(g) $	1 stable SS
$ g<4 $	$ \nu>N_2(g) $	1 unstable SS, 2 stable ASs
$ 4<g<16 $	$ \nu<N_1(g) $	1 stable SS
	$ N_1(g)<\nu<N_2(g) $	1 stable SS, 2 stable ASs and 2 unstable ASs
	$ \nu>N_2(g) $	1 unstable SS, 2 stable ASs
$ g>16 $	$ \nu<N_1(g) $	1 stable SS
	$ N_1(g)<\nu<N_2(g) $	1 stable SS, 2 stable ASs, 2 unstable ASs
	$ N_2(g)<\nu<N_{3}(g) $	1 unstable SS and 2 stable ASs
	$ N_3(g)<\nu<N_{4}(g) $	3 unstable SSs and 2 stable ASs
	$ \nu>N_{4}(g) $	1 unstable SS and 2 stable ASs

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Table 2. Number of steady states of system (7) and their types

$g$	$\nu$	number and type	$~$total$~\, $
$0<g\leq 4$	$0<\nu\leq{g}/{2}+2$	1 SS (stable node)	1
	$\nu>{g}/{2}+2$	1 SS (saddle)	3
	$\nu>{g}/{2}+2$	2 ASs (stable nodes)	3
$4<g<16$	$0<\nu<2\sqrt g$	1 SS (stable node)	1
	$\nu=2\sqrt g$	1 SS (stable node)	3
	$\nu=2\sqrt g$	2 ASs(saddle-nodes)	3
	$2\sqrt g<\nu<{g}/{2}+2$	1 SS (stable node)	5
		2 ASs (stable nodes)
		2 ASs (saddle)
	$\nu\geq{g}/{2}+2$	1 SS (saddle)	3
	$\nu\geq{g}/{2}+2$	2 ASs (stable nodes)	3
$g=16$	$0<\nu<8$	1 SS (stable node)	1
	$\nu=8$	1 SS (stable node)	3
	$\nu=8$	2 ASs (saddle-nodes)	3
	$8<\nu<10$	1 SS (stable node)	5
		2 ASs (stable nodes)
		2 ASs (saddle)
	$\nu\geq 10$	1 SS (saddle)	3
	$\nu\geq 10$	2 ASs (stable nodes)	3
$16<g\leq 8+8\sqrt2$	$0<\nu<2\sqrt g$	1 SS (stable node)	1
	$\nu=2\sqrt g$	1 SS (stable node)	3
	$\nu=2\sqrt g$	2 ASs (saddle-nodes)	3
	$2\sqrt g<\nu<{g}/{2}+2$	1 SS (stable node)	5
		2 ASs (stable nodes)
		2 ASs (saddle)
	${g}/{2}+2\leq \nu<N_1^*(g)$	1 SS (saddle)	3
	${g}/{2}+2\leq \nu<N_1^*(g)$	2 ASs (stable nodes)	3
	$\nu=N^*_1(g)$	1 SS (saddle)	4
		1 SS (saddle-node)
		2 ASs (stable nodes)
	$N^_1(g)<\nu<N^_2(g)$	1 SS (unstable node)	5
		2 SSs (saddle)
		2 ASs (stable nodes)
	$\nu=N^*_2(g)$	1 SS (saddle-node)	4
		1 SS (saddle)
		2 ASs (stable nodes)
	$\nu>N^*_2(g)$	1 SS (saddle)	3
	$\nu>N^*_2(g)$	2 ASs (stable nodes)	3
$ g>8+8\sqrt2 $	$ 0<\nu<2\sqrt g $	1 SS (stable node)	1
	$ \nu=2\sqrt g $	1 SS (stable node)	3
	$ \nu=2\sqrt g $	2 ASs (saddle-nodes)	3
	$ 2\sqrt g<\nu<N^*_1(g) $	1 SS (stable node)	5
		2 ASs (stable nodes)
		2 ASs (saddle)
	$ \nu=N^*_1(g) $	1 SS (stable node)	6
		1 SS (saddle-node)
		2 ASs (stable nodes)
		2 ASs (saddle)
	$ N^*_1(g)<\nu<{g}/{2}+2 $	1 SS (unstable node)	7
		1 SS (saddle)
		1 SS (stable node)
		2 ASs (stable node)
		2 ASs (saddle)
	$ {g}/{2}+2\leq \nu<N^*_2(g) $	1 SS (unstable node)	5
		2 SSs (saddle)
		2 ASs (stable nodes)
	$ \nu=N^*_2(g) $	1 SS (saddle-node)	4
		1 SS (saddle)
		2 ASs (stable nodes)
	$ \nu>N^*_2(g) $	1 SS (saddle)	3
	$ \nu>N^*_2(g) $	2 ASs (stable nodes)	3

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Table 3. Nonhyperbolic cases, where $ s_\pm $ and $ x_*^\pm $ are defined in (13) and (35)

Label	$ g $, $ \nu $	degenerate steady-states and coordinates
NH1	$ g>16 $, $ \nu=N_1^*(g) $	1 SS(saddle-node) $ (s_+, s_+) $
NH2	$ g>16 $, $ \nu=N_2^*(g) $	1 SS(saddle-node) $ (s_-, s_-) $
NH3	$ g=16 $, $ \nu=6\sqrt3 $	1 SS(saddle) $ (\sqrt3, \sqrt3) $
NH4	$ g>4 $, $ \nu=N_1(g) $	2 ASs(saddle-nodes) $ (x_^+, x_^-) $ and $ (x_^-, x_^+) $
NH5	$ g>4 $, $ \nu=N_2(g) $	1 SS(saddle) $ (1, 1) $
NH5	$ 0<g<4 $, $ \nu=N_2(g) $	1 SS(stable node) $ (1, 1) $
NH6	$ g=4 $, $ \nu=4 $	1 SS(stable node) $ (1, 1) $

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