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A Mathematical Model to describe the herd behaviour considering group defense

R. A. de Assis

R. A. de Assis

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de Assis, R. A.
, 
R. Pazim

R. Pazim

Instituto de Ciências Naturais Humanas e Sociais, Universidade Federal de Mato GrossoSinop, Brazil
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Pazim, R.
, 
M. C. Malavazi

M. C. Malavazi

Instituto de Ciências Naturais Humanas e Sociais, Universidade Federal de Mato GrossoSinop, Brazil
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Malavazi, M. C.
, 
P. P. da C. Petry

P. P. da C. Petry

Faculdade de Ciências Exatas e Tecnológicas, Universidade do Estado de Mato GrossoSinop, Brazil
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Petry, P. P. da C.
, 
L. M. E. de Assis

L. M. E. de Assis

Faculdade de Ciências Exatas e Tecnológicas, Universidade do Estado de Mato GrossoSinop, Brazil
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de Assis, L. M. E.
 and 
E. Venturino

E. Venturino

Dipartimento di Matematica “Giuseppe Peano”, Università di TorinoItaly
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Venturino, E.
  
Jan 31, 2020
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Published Online: Jan 31, 2020
Page range: 11 - 24
Received: Mar 09, 2019
Accepted: Sep 23, 2019
DOI: https://doi.org/10.2478/amns.2020.1.00002
Keywords
© 2020 R. A. de Assis et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Response functions g(R) and f(R) of population R for the parameters a2 = 1, h˜=20\tilde h = 20 and a1=120{a_1} = {1 \over {\sqrt {20} }} .
Response functions g(R) and f(R) of population R for the parameters a2 = 1, h˜=20\tilde h = 20 and a1=120{a_1} = {1 \over {\sqrt {20} }} .

Fig. 2

(a) The transcritical and Hopf bifurcation curves, respectively, T={(h,μ),μ=μ†=1h+1}T = \{ {(h,\mu ),\mu = {\mu ^\dagger } = {1 \over {\sqrt {h + 1} }}} \} and H={(h,μ),μ=μ*=1−2h3(1+h)}H = \{ {(h,\mu ),\mu = {\mu ^*} = {{1 - 2h} \over {\sqrt {3(1 + h} )}}} \} , determine three regions in the parameter plane (h, μ). The equilibrium point E3 is unstable in (1) “saddle” and (3) “focus or node”, and stable in (2) “focus or node”. The equilibrium point E2 is stable in (1) “node” and unstable in (2) and (3) “saddle”. (b) The curve C = {R3 = h} determines two regions in the parameter plane (h, μ). Namely, (4) region of group defense and (5) region without group defense.
(a) The transcritical and Hopf bifurcation curves, respectively, T={(h,μ),μ=μ†=1h+1}T = \{ {(h,\mu ),\mu = {\mu ^\dagger } = {1 \over {\sqrt {h + 1} }}} \} and H={(h,μ),μ=μ*=1−2h3(1+h)}H = \{ {(h,\mu ),\mu = {\mu ^*} = {{1 - 2h} \over {\sqrt {3(1 + h} )}}} \} , determine three regions in the parameter plane (h, μ). The equilibrium point E3 is unstable in (1) “saddle” and (3) “focus or node”, and stable in (2) “focus or node”. The equilibrium point E2 is stable in (1) “node” and unstable in (2) and (3) “saddle”. (b) The curve C = {R3 = h} determines two regions in the parameter plane (h, μ). Namely, (4) region of group defense and (5) region without group defense.

Fig. 3

The transcritical bifurcation between the equilibrium points E2 and E3 as function of the bifurcation parameter μ. a) For μ = 1.05, λ = 0.7 and h = 0.15, E2 is a stable node and E3 is a saddle. b) E3 = E2 is a nonhyperbolic equilibrium for μ = μ† ≈ 0.932, λ = 0.7 and h = 0.15. c) For μ = 0.88, λ = 0.7 and h = 0.15, E2 is a saddle and E3 is a stable node.
The transcritical bifurcation between the equilibrium points E2 and E3 as function of the bifurcation parameter μ. a) For μ = 1.05, λ = 0.7 and h = 0.15, E2 is a stable node and E3 is a saddle. b) E3 = E2 is a nonhyperbolic equilibrium for μ = μ† ≈ 0.932, λ = 0.7 and h = 0.15. c) For μ = 0.88, λ = 0.7 and h = 0.15, E2 is a saddle and E3 is a stable node.

Fig. 4

Taking μ as a variation parameter a limit cycle arises from equilibrium E3R, indicating the onset of a Hopf bifurcation. a) For μ = 0.65 > μ*, λ = 0.7 and h = 0.15, E3 is a stable focus. b) For μ = μ* ≈ 0.376, λ = 0.7 and h = 0.15, E3 is a weak stable focus. c) For μ < μ*, λ = 0.7 and h = 0.15 E3 is an unstable focus.
Taking μ as a variation parameter a limit cycle arises from equilibrium E3R, indicating the onset of a Hopf bifurcation. a) For μ = 0.65 > μ*, λ = 0.7 and h = 0.15, E3 is a stable focus. b) For μ = μ* ≈ 0.376, λ = 0.7 and h = 0.15, E3 is a weak stable focus. c) For μ < μ*, λ = 0.7 and h = 0.15 E3 is an unstable focus.

Behaviour and conditions of feasibility and stability of equilibria for the model (4), with 0 < h < 1, μ > 0, and μ* and μ† respectively defined in (11), (10)_ Note that if μ > μ†, E3 is unfeasible, having negative coordinates, although without direct biological meaning, this equilibrium collides with E2 when it becomes feasible, going through a transcritical bifurcation_

EquilibriaFeasibilityStability
E1alwaysunstable (saddle)
E2alwaysunstable (saddle) if μ < μ†
stable if μ > μ†
E30 ≤ μ ≤ μ†unstable (saddle) if μ > μ†
stable (node/focus) if max{0, μ*} < μ < μ†
unstable (focus) if 0 < μ < μ*
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