| Citation: |
Haiyan Xu, Zhigui Lin, Carlos Alberto Santos. PERSISTENCE, EXTINCTION AND BLOWUP IN A GENERALIZED LOGISTIC MODEL WITH IMPULSES AND REGIONAL EVOLUTION[J]. Journal of Applied Analysis & Computation, 2022, 12(5): 1922-1944. doi: 10.11948/20210393
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PERSISTENCE, EXTINCTION AND BLOWUP IN A GENERALIZED LOGISTIC MODEL WITH IMPULSES AND REGIONAL EVOLUTION
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Abstract
To explore the impacts of regional evolution and impulses on the persistence or extinction of species, a generalized logistic model with impulses in an evolving domain is proposed and researched. Firstly, the ecological reproduction index, which is regarded as a threshold value, is introduced and characterized. Secondly, in the case of monotone or non-monotone impulsive function, the asymptotic behavior of population is fully investigated and the sufficient conditions for the solution to persist, be extinct or blow up are given. Finally, numerical simulations results indicate that whatever impulse is, larger periodic evolution rates are more favorable for species. However, impulsive harvesting has a negative impact on persistence of species, while birth pulse admits a positive impact and even results in blowup.
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References
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Haiyan Xu, Zhigui Lin, Carlos Alberto Santos. PERSISTENCE, EXTINCTION AND BLOWUP IN A GENERALIZED LOGISTIC MODEL WITH IMPULSES AND REGIONAL EVOLUTION[J]. Journal of Applied Analysis & Computation, 2022, 12(5): 1922-1944. doi: 10.11948/20210393
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Figure 1. For a smaller evolution rate
$ \rho_1(t)=e^{-0.1(1-\cos{\pi{t}})} $ , we obtain$ {\mathcal{R}}_0\approx{0.993}<1 $ . Graphs (a)-(c) show that$ u(t, x) $ gradually tends to zero with time$ t $ . It can be seen from Graph (c) that the impulse occurs at each time$ T=2 $ . -
Figure 2. For bigger evolution rate
$ \rho_2(t)=e^{-0.1(1-\cos{\pi{t}})} $ , we obtain$ {\mathcal{R}}_0\approx{1.496}>1 $ . is calculated. Graphs (a)-(c) characterize the population density stabilizes to a equilibrium state in a periodic evolving domain. -
Figure 3. A numerical simulation to the graph of
$ u(t, x) $ without impulses by choosing$ \rho_1(t)=e^{-0.1(1-\cos{\pi{t}})} $ .$ {\mathcal{R}}_0\approx{0.973} <1 $ , Graphs (a)-(c) depict that the species finally tends to vanish. -
Figure 4. For the smaller evolution rate, Beverton-Holt impulse with
$ m=8, a=10 $ , we acquire$ {\mathcal{R}}_0\approx{0.892}<1 $ .Graphs (a)-(c) describe that the population density decays to zero. It can be seen from Graph (c) that the impulse arises at each time$ T=2 $ . -
Figure 5. Without impulses, a numerical approximation to the graph of
$ u(t, x) $ . By choosing$ \rho_2(t)=e^{0.1(1-\cos{\pi{t}})} $ ,$ {\mathcal{R}}_0\approx{1.451}>1 $ . Graphs (a)-(c) show that the population gradually tends to a steady state. -
Figure 6.
$ \rho_2(t)=e^{0.1(1-\cos{\pi{t}})} $ and Beverton-Holt function with$ m=4, a=10 $ are given, such that$ {\mathcal{R}}_0\approx{0.934}<1 $ . Graphs (a)-(c) portray that population density$ u(t, x) $ decays ultimately to zero. -
Figure 7.
$ \rho_2(t)=e^{0.1(1-\cos{\pi{t}})} $ , Ricker function with parameter values$ r=0.05 $ and$ b=8 $ are given such that$ {\mathcal{R}}_0\approx{1.496}>1 $ . Graphs (a)-(c) characterize that the population stabilizes to a smaller positive steady state. -
Figure 8.
$ \rho_2(t)=e^{0.1(1-\cos{\pi{t}})} $ and big initial value$ v_0(y)=14\sin{x}+10\sin{3x} $ . Graphs (a) and (b) with$ c=0.4(<1) $ characterizes the solution tends to zero, while$ c=1.5(>1) $ in Graphs (c) and (d) means$ u(t, x) $ still blow up. -
Figure 9. Graphs (a) and (b) means the population will be extinct with
$ c=0.47(<1) $ , while Graphs (c)-(d) indicate the solution will blow up with$ c=1.4(>1) $ . -
Figure 10. Graph (a) means
$ u(t, x) $ decays to zero with$ c=0.5(c<1) $ , Graph (b) characterizes the population stabilizes to a steady state with$ c=1.07(>1) $ , and Graph (c) indicates that the solution will blow up when$ c=1.4(>1) $ .