Article
Kyungpook Mathematical Journal 2020; 60(2): 335-347
Published online June 30, 2020
Copyright © Kyungpook Mathematical Journal.
Bifurcation Analysis of a Spatiotemporal Parasite-host System
Hunki Baek
Department of Mathematics Education, Daegu Catholic University, Gyeongsan, Gyeongbuk, 38430, Republic of Korea
e-mail : hkbaek@cu.ac.kr
Received: April 25, 2018; Accepted: March 7, 2020
Abstract
In this paper, we take into account a parasite-host system with reaction-diffusion. Firstly, we derive conditions for Hopf, Turing, and wave bifurcations of the system in the spatial domain by means of linear stability and bifurcation analysis. Secondly, we display numerical simulations in order to investigate Turing pattern formation. In fact, the numerical simulation discloses that typical Turing patterns, such as spotted, spot-stripelike mixtures and stripelike patterns, can be formed. In this study, we show that typical Turing patterns, which are well known in predator-prey systems ([
Keywords: a spatiotemporal parasite-host system,Turing bifurcation, Hopf bifurcation, wave bifurcation, pattern formation
1. Introduction
It is widely known that parasites play an important role in reducing host density and in extinctionizing the host population in some cases([2, 8]). In order to describe such phenomena by means of mathematical models, Ebert et al. [8] suggested the following microparasite model
where
From simple local stability analysis, we know that system (
After carefully investigating the infection term
where
Now, for simplicity, we nondimensionalizes the system (
and dropping the bar notation, then we have the following system:
where
Since lim(
Although system (
where
which means that no external input is imposed from outside.
The purposes of this paper are to investigate three bifurcation phenomena, Hopf, Turing and wave bifurcations, of system (
2. Linear Stability Analysis and Hopf Bifurcation of System (1.4 )
In order to investigate bifurcation phenomena of system (
-
(i)
E 0 = (0, 0), -
(ii)
E 1 = (1 −δ , 0) if 0 <δ < 1, -
(iii)
E * = (S *,I *) ifh >δ +r andbh + (δ +r )(r + 1) > (b +h )(δ +r ),
where
Here
From [12], the stability of the equilibrium
Proposition 2.1
-
(a)
If h ≤δ +r and δ ≥ 1,then E 0is globally asymptotically stable. -
(b)
If h >r + 1and bh + (δ +r )(r + 1) < (b +h )(δ +r ),then E 0is globally stable.
Now we consider the stability of the equilibrium
Proposition 2.2
-
(a)
If h <δ +r, then E 1is globally asymptotically stable. -
(b)
If h >δ +r, then E 1is a saddle point. -
(c)
If h =δ +r, then E 1is a saddle node.
Parts (a) and (b) follow from [12] and the variational matrix defined by
Now, let
where
where
In the following proposition, we mention the local stability of the equilibrium
Proposition 2.3
-
(a)
If δ +r <h ≤r + 1,then E *is globally asymptotically stable. -
(b)
If h >r + 1and bh + (δ +r )(r + 1) > (b +h )(δ +r ),then E *is globally stable.
Now we can investigate Hopf-bifurcation phenomenon around the equilibrium
Theorem 2.4
Consider the characteristic equation at the equilibrium point
where tr(
From elementary differentiation of (
where
Thus it follows from (
3. Turing and Wave Bifurcations of System (1.5 )
In this section we consider the spatiotemporal parasite-host system (
where
where
where
The solutions of
The reaction-diffusion systems have led to the characterization of three basic types of symmetry-breaking bifurcations-Hopf, Turing and wave bifurcation, which are responsible for the emergence of spatiotemporal patterns [1, 5, 6, 10, 14, 15, 18, 20, 19, 25, 23, 22, 24]).
3.1. Turing Bifurcation
Turing bifurcation(or called Turing instability) is a phenomenon that causes certain reaction-diffusion system to lead to spontaneous stationary configuration. It is why Turing instability is often called
In fact, Turing instability sets in when at leat one of the solutions of
Thus we can get the conditions for Turing bifurcation in the following theorem.
From the hypotheses and Proposition, we know that there exists the positive non-spatial steady state
where
Thus the minimum of
By substituting
Remark 3.2
We can find out the critical value of bifurcation parameter
At the Turing threshold
3.2. Wave Bifurcation
The wave instability caused by the wave bifurcation plays an important part in pattern formations in many areas [20, 19, 26]. Similar to the Hopf bifurcation the wave bifurcation take places when a pair of imaginary eigenvalues across the real axis from the negative the positive side for
Theorem 3.3
In order that system (
Thus, from elementary calculation, the critical value of wave bifurcation parameter
where
Remark 3.4
In fact, at the wave threshold
4. Numerical Simulation: Turing pattern
In this section, we display numerical simulations of the spatiotemporal parasite-host system (
For this, we adopt a finite difference numerical method for the spatial derivatives, an explicit Euler method for the time integration and no-flux boundary condition for the boundary condition. Also we use a finite nondimensional domain of [0, 200] × [0, 200], and the space step Δ
In order to illustrate Turing patterns numerically, first, we fix parameter values in system (
If we take the parameter value
Thus from these figures we can infer that Turing bifurcation causes three typical Turing patterns such as spotted, spot-stripe mixture, and stripelike patterns of the uninfected population in system (
Figures
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