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Journal of Biological Dynamics Volume 13, 2019 - Issue 1
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Articles

Bifurcation analysis and chaos control for a plant–herbivore model with weak predator functional response

Qamar DinDepartment of Mathematics, University of the Poonch Rawalakot, Rawalakot, PakistanCorrespondence[email protected]
[email protected]
ORCID Iconhttps://orcid.org/0000-0002-0999-7404
ORCID Icon,
M. Sajjad ShabbirDepartment of Mathematics, Air University Islamabad, Islamabad, PakistanCorrespondence[email protected]
,
M. Asif KhanDepartment of Mathematics, University of the Poonch Rawalakot, Rawalakot, PakistanCorrespondence[email protected]
&
Khalil AhmadDepartment of Mathematics, Air University Islamabad, Islamabad, PakistanCorrespondence[email protected]
Pages 481-501 | Received 24 Feb 2019, Accepted 27 Jun 2019, Published online: 09 Jul 2019

Figures & data

Figure 1. Existence region (red) for P∗ at β=0.2.

Figure 1. Existence region (red) for P∗ at β=0.2.

Figure 2. Topological classification of P0 for α∈[0,2] and γ∈[0,2].

Figure 2. Topological classification of P0 for α∈[0,2] and γ∈[0,2].

Figure 3. Topological classification of P1 for α∈[0,1], β∈[0,200] and γ=0.995.

Figure 3. Topological classification of P1 for α∈[0,1], β∈[0,200] and γ=0.995.

Figure 4. Topological classification of P∗ for 0<α<1, 0<γ<1 and β=0.1.

Figure 4. Topological classification of P∗ for 0<α<1, 0<γ<1 and β=0.1.

Figure 5. Bifurcation diagrams and MLE for system (Equation1) with α=0.99, β=0.008, γ∈[0.2,0.7] and (x0,y0)=(1.25,0.001): (a) bifurcation diagram for xn, (b) bifurcation diagram for yn and (c) MLE.

Figure 5. Bifurcation diagrams and MLE for system (Equation1(1) xn+1=xnα(1+yn2)+βxn,yn+1=γyn(1+xn),(1) ) with α=0.99, β=0.008, γ∈[0.2,0.7] and (x0,y0)=(1.25,0.001): (a) bifurcation diagram for xn, (b) bifurcation diagram for yn and (c) MLE.

Figure 6. Bifurcation diagrams and MLE for system (Equation1) with α=0.2, β=0.5, γ∈[0.45,0.95] and (x0,y0)=(0.78,1.426): (a) bifurcation diagram for xn, (b) bifurcation diagram for yn and (c) MLE.

Figure 6. Bifurcation diagrams and MLE for system (Equation1(1) xn+1=xnα(1+yn2)+βxn,yn+1=γyn(1+xn),(1) ) with α=0.2, β=0.5, γ∈[0.45,0.95] and (x0,y0)=(0.78,1.426): (a) bifurcation diagram for xn, (b) bifurcation diagram for yn and (c) MLE.

Figure 7. Phase portraits of system (Equation1) for α=0.2, β=0.5, x0=0.78,y0=1.426 and with different values of γ: (a) phase portrait for γ=0.573, (b) phase portrait for γ=0.576923, (c) phase portrait for γ=0.58 and (d) phase portrait for γ=0.65.

Figure 7. Phase portraits of system (Equation1(1) xn+1=xnα(1+yn2)+βxn,yn+1=γyn(1+xn),(1) ) for α=0.2, β=0.5, x0=0.78,y0=1.426 and with different values of γ: (a) phase portrait for γ=0.573, (b) phase portrait for γ=0.576923, (c) phase portrait for γ=0.58 and (d) phase portrait for γ=0.65.

Figure 8. Bounded stability region for system (Equation24).

Figure 8. Bounded stability region for system (Equation24(24) xn+1=xn0.2(1+yn2)+0.5xn,yn+1=(0.95−k1(xn−0.0526316)−k2(yn−1.96683))yn(1+xn).(24) ).

Figure 9. Plots for system (Equation25) with κ=0.33 and (x0,y0)=(0.0526316,1.96683): (a) plot for xn, (b) plot for yn and (c) phase portrait.

Figure 9. Plots for system (Equation25(25) xn+1=κ(xn0.2(1+yn2)+0.5xn)+(1−κ)xn,yn+1=κ(0.95yn(1+xn))+(1−κ)yn.(25) ) with κ=0.33 and (x0,y0)=(0.0526316,1.96683): (a) plot for xn, (b) plot for yn and (c) phase portrait.

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