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A Mathematical Model to describe the herd behaviour considering group defense

R. A. de Assis
R. A. de Assis
, 
R. Pazim
R. Pazim
, 
M. C. Malavazi
M. C. Malavazi
, 
P. P. da C. Petry
P. P. da C. Petry
, 
L. M. E. de Assis
L. M. E. de Assis
 und 
E. Venturino
E. Venturino
   | 31. Jan. 2020
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Online veröffentlicht: 31. Jan. 2020
Seitenbereich: 11 - 24
Eingereicht: 09. März 2019
Akzeptiert: 23. Sept. 2019
DOI: https://doi.org/10.2478/amns.2020.1.00002
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© 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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