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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

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 ([7, 18, 25]), can be observed in a parasite-host system as well.

Keywords: a spatiotemporal parasite-host system,Turing bifurcation, Hopf bifurcation, wave bifurcation, pattern formation

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

{dSdt=a(S+bI)(1-c(S+I))-eS-βSI,dIdt=-(e+α)I+βSI,

where S and I represent the densities of uninfected and infected hosts, respectively. The constant a is the maximum per capita birth rate of uninfected hosts, b(0 ≤ b ≤ 1) is the relative fecundity of an infected host, c measures the per capita density-dependent reduction in birth rate, e is the parasite-independent host background mortality, and α is the parasite-induced excess death rate and β is the constant infection rate.

From simple local stability analysis, we know that system (1.1) satisfying a > d has always the saddle equilibrium O(0, 0), which implies that extinction of host is impossible under a > d. In addition, in [12], they have shown that system (1.1) predicts the existence of a globally attractive positive steady state. Thus it is not a suitable model to explain the extinction situation of the uninfected host.

After carefully investigating the infection term βSI, Hwang and Kuang [12] obtained the following system by replacing the mass action incidence function βSI with a standard incidence function βSIS+I.

{dSdt=a(S+bI)(1-c(S+I))-eS-βSIS+I,dIdt=-(e+α)I+βSIS+I,

where β represents the maximum number of infections that an infected host can cause per unit time. In [12, 21], the authors have studied about various dynamical behaviors of the revised system (1.2) including the fact that the host extinction and reduction dynamics can be exhibited.

Now, for simplicity, we nondimensionalizes the system (1.2) with the following scaling

t¯=at,S¯=cS,I¯=cI,

and dropping the bar notation, then we have the following system:

{dSdt=(S+bI)(1-S-I)-δS-hSIS+IP(S,I),dIdt=-(δ+r)I+hSIS+IQ(S,I),

where h=βa,δ=da,r=αa.

Since lim(S,I)→(0,0)P(S, I) = lim(S,I)→(0,0)Q(S, I) = 0 we define that P(0, 0) = Q(0, 0) = 0. With this assumption, we know that both P and Q are continuous on the closure of +2 and C1 smooth in +2 where +2={(x,y)x>0,y>0}. Thus, standard arguments yields that the solutions of system (1.4) are positive, bounded and defined on [0, ∞).

Although system (1.4) has been investigated by researchers [12, 16, 21] about its dynamics as an ordinary differential systems, one cannot infer any information about spatial distributions from such systems. Thus, it is needed to investigate spatial system. In fact, recently, many researchers have investigated spatiotemporal pattern formations and bifurcation analysis of spatiotemporal predator-prey systems [1, 3, 4, 5, 10, 15, 18, 20, 19, 25, 23, 22, 24, 26]. Thus, in this paper, we will take into account the following a parasite-host system with reaction-diffusion:

{dSdt=D12S+(S+bI)(1-S-I)-δS-hSIS+Iin Ω,dIdt=D22I-(δ+r)I+hSIS+I,in Ω,

where D1 and D2 are diffusion coefficients, 2=2S2+2I2 is the usual Laplacian in two-dimensional space Ω and S, I stand for the space. Throughout the paper, we assume that system (1.5) has the Neumann boundary conditions as follows:

Sn=In=0on Ω,

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 (1.5) and to give numerical simulations in order to observe Turing patterns caused by Turing bifurcation. In fact, in section, we establish the conditions for Hopf, Turing and wave bifurcations by using linear stability theory and bifurcation analysis and, in section 3, we analyze numerically typical Turing patterns via numerical simulations.

In order to investigate bifurcation phenomena of system (1.5), first we have to consider the nonspatial system (1.4) of system (1.5). In fact, by setting P(S, I) = 0, Q(S, I) = 0 in system (1.4), we figure out that the nonspatial system (1.4) has at most three nonnegative equilibria as follows;

  • (i) E0 = (0, 0),

  • (ii)E1 = (1 − δ, 0) if 0 < δ < 1,

  • (iii)E* = (S*, I*) if h > δ + r and bh + (δ + r)(r + 1) > (b + h)(δ + r),

where

S*=δ+rh-δ-rI*,I*=(h-δ-r)(b(h-δ-r)-(δ+r)(h-r-1))h(bh+(1-b)(δ+r)).

Here E1 represents that there are no infected hosts and E* implies that host is infected chronically. Since the local stability of equilibria E0, E1 and E* is determined by the eigenvalues of the variational matrix, we need to consider the variational matrix of system (1.5) at a point (x, y) ∈ Ω given by

J(x,y)=(PSPIQSQI)(x,y)=(1-δ-2x-(1+b)y-hy2(x+y)2b-(1+b)x-2by-hx2(x+y)2hy2(x+y)2-(δ+r)+hx2(x+y)2).

From [12], the stability of the equilibrium E0 can be obtained as shown in the following proposition.

Proposition 2.1

For system (1.4), the following statements are true.

  • (a) If hδ + r and δ ≥ 1, then E0is globally asymptotically stable.

  • (b) If h > r + 1 and bh + (δ + r)(r + 1) < (b + h)(δ + r), then E0is globally stable.

Now we consider the stability of the equilibrium E1 in the following proposition.

Proposition 2.2

Let 0 < δ < 1 in system (1.4). Then the following statements are true.

  • (a) If h < δ + r, then E1is globally asymptotically stable.

  • (b) If h > δ + r, then E1is a saddle point.

  • (c) If h = δ + r, then E1is a saddle node.

Proof

Parts (a) and (b) follow from [12] and the variational matrix defined by

J(E1)=(δ-1b0h-(δ+r)).

Now, let h = δ + r for the part (c). Then the trace of the matrix J(E1) is not zero, one of the eigenvalues of the matrix J(E1) is zero and the other is nonzero. Thus in order to determine the dynamics of system (1.4) in the neighborhood of the equilibrium E1, we transform the equilibrium E1 to the origin and then expand the right hand side of system (1.4) as a Taylor series. Then system (1.4) can be written as

{dSdt=(δ-1)S+(bδ-r-1)I-S2+b(1-δ)-δ-rδ-1I2-(b+1)SI+P1(S,I),dIdt=δ+rδ-1I2+Q1(S,I),

where P1(S, I) and Q1(S, I) are C functions of order at least three in (S, I). Let S = x − (r − 1)y, I = (δ − 1)y and τ = (δ − 1)t, then system (2.4) can be transformed into the following system:

{dxdτ=x-1δ-1x2+(b(δ+1)-δ-2r-1)δ-rxy+((-1+b)δ2+(b-2r-1)r-δ(b2+3r+1-2b(r+1)))δ-1y2+P2(x,y),dydτ=(δ+r)δ-1y2+Q2(x,y),

where P2(x, y) and Q2(x, y) are C functions of order at least three in (x, y). Now using Theorem 7.1 of Chapter 2 in [27], we can obtain that E1 is a saddle node.

In the following proposition, we mention the local stability of the equilibrium E* obtained from [12].

Proposition 2.3

For system (1.4), the following statements are true.

  • (a) If δ +r < hr + 1, then E*is globally asymptotically stable.

  • (b) If h > r + 1 and bh + (δ + r)(r + 1) > (b + h)(δ + r), then E*is globally stable.

Now we can investigate Hopf-bifurcation phenomenon around the equilibrium E* in the following theorem.

Theorem 2.4

System (1.4) can have a Hopf-bifurcation around E*at b = bH if

Γ1-Γ2-Γ30

where bH satisfies the following equality of b

1-2δ+h-r-2hθ+1-γ=0

and

θ=δ+rh-δ-r,γ=(-δ+h-r)(-(-1+h-r)(δ+r)-b(δ-h+r))(1+b+2θ)h(bh-(-1+b)(δ+r)),Γ1=(h-r)(δ+r)(δ-h+r)(δ-bHδ+h+bHh+r-bHr)h(δ-bHδ+bHh+r-bHr)2,Γ2=2(h-r)(δ+r)(δ-h+r)(δ+r+(δ-h+r)θ)Γ3((-1+bH)δ-bHh+(-1+bH)r)2(1+θ)2,Γ3=(h-δ-r)(bH(h-δ-r)-(δ+r)(h-r-1))h(bHh+(1-bH)(δ+r)).
Proof

Consider the characteristic equation at the equilibrium point E* as follows:

λ2-tr(J(E*))λ+det(J(E*))=0

where tr(J(E*)) and det(J(E*)) is the trace and the determinant of the of the variational matrix J(E*) of the point E*, respectively. If tr(J(E*)) = 0 at b = bH, then both the eigenvalues, the solutions of the equation (2.7), become purely imaginary under the condition det(J(E*)) > 0. Now, replacing λ = λ1 + 2 into the equation (2.7) and then from separating real and imaginary parts we can get the followings:

(λ12-λ22)-tr(J(E*))λ1+det(J(E*))=0,2λ1λ2-tr(J(E*))λ2=0.

From elementary differentiation of (2.8b) with respect to b and considering λ1 = 0, we get

dλ1db|b=bH=Γ1-Γ2-Γ3

where

θ=δ+rh-δ-r,Γ1=(h-r)(δ+r)(δ-h+r)(δ-bHδ+h+bHh+r-bHr)h(δ-bHδ+bHh+r-bHr)2,Γ2=2(h-r)(δ+r)(δ-h+r)(δ+r+(δ-h+r)θ)Γ3((-1+bH)δ-bHh+(-1+bH)r)2(1+θ)2,Γ3=(h-δ-r)(bH(h-δ-r)-(δ+r)(h-r-1))h(bHh+(1-bH)(δ+r)).

Thus it follows from (2.6) that dλ1db|b=bH0. Therefore, by Poincaré-Andronov-Hopf Theorem [11], system (1.4) goes through a Hopf-bifurcation at b = bH around E*.

In this section we consider the spatiotemporal parasite-host system (1.5) in order to look into Turing and wave bifurcations. For this, we will perform a linear stability analysis for system (1.5) around the nontrivial stationary state (S*, I*) by linearizing the dynamical system (1.5) around the spatially homogeneous fixed point (S*, I*) for small space- and time-dependent fluctuations and expand them in Fourier space. Now, let

S(x,t)~S*eλteik.x,I(x,t)~I*eλteik.x,

where x⃗ = (S, I), λ is the growth rate of perturbation in time t and k⃗ = (kS, kI) is the wave number vector. Let k·k=kS2+kI2k2. Then we can obtain the corresponding characteristic equation as follows:

Jk-λE=0,

where Jk = J(E*) − k2D, E is the identity matrix and D = diag(D1, D2) is the diffusion matrix and J(E*) is given in (2.2) as

J(E*)=(PSPIQSQI)(S*,I*)(PSPIQSQI).

Equation (3.2) can be solved, yielding the characteristic polynomial

λ2-tr(Jk)λ+det(Jk)=0,

where

tr(Jk)=PS+QI-k2(D1+D2)anddet(Jk)=PSQI-PIQS-k2(D1QI+D2PS)+k4D1D2.

The solutions of equation (3.4) yield the dispersion relation

λk±=12(tr(Jk)=tr(Jk)2-4det(Jk)).

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 diffusion-driven instability. Turing instability is not dependent on the geometry of the system but only on the reaction rates and diffusion. It can occur only when the inhibitor(S) diffuses faster than the activator(I) [15, 25, 23].

In fact, Turing instability sets in when at leat one of the solutions of equation (3.4) crosses the imaginary axis. In other words, the spatially homogeneous steady state will become unstable due to heterogeneous perturbation when at least one solution of equation (3.4) is positive. For the reason, at least one out of the following two inequalities is violated to occur the Turing instability phenomenon:

tr(Jk)=PS+QI-(D1+D2)k2<0,det(Jk)=D1D2k4-(D1QI+D2PS)k2+PSQI-PIQS>0.

Thus we can get the conditions for Turing bifurcation in the following theorem.

Theorem 3.1

Suppose that h > δ + r and bh + (δ + r)(r + 1) > (b + h)(δ + r) hold. Then Turing bifurcation occurs if

-14D1D2(D1(δ-h+2η-η2h+r)+D2(δ-1+η2h+θ2θ1(2+ηh(b-1))))2+θ2θ1(2(δ-h)+(5-b)η+2r+(b-1)ηh(δ+3η+r))+(δ-1)(δ+2η+r-h)+η2h(1-b+r)<0

is satisfied. Here η = hδr, θ1 = + δ + r, θ2 = θ1 − (hη)(hr).

Proof

From the hypotheses and Proposition, we know that there exists the positive non-spatial steady state E* which is stable. Thus tr(J(E*)) = PS + QI < 0 and det(J(E*)) = PSQIPIQS > 0 are satisfied. It is seen from these facts that the first condition in (3.7) always holds. Hence we are left only one for the instability condition, i.e., T(k2) ≡ det(Jk) < 0. Elementary calculations yield that

T(k2)=(D1(δ-h+2η-η2h+r)+D2(δ-1+η2h+θ2θ1(2+ηh(b-1))))k2+θ2θ1(2(δ-h)+(5-b)η+2r+(b-1)ηh(δ+3η+r))+(δ-1)(δ+2η+r-h)+η2h(1-b+r)+D1D2k4

where

η=h-δ-r,θ1=bη+δ+r,θ2=θ1-(h-η)(h-r).

Thus the minimum of T(k2) occurs at the critical wavenumber kT2, where

kT2=-12D1D2(D1(δ-h+2η-η2h+r)+D2(δ-1+η2h+θ2θ1(2+ηh(b-1)))),η=h-δ-r,θ1=bη+δ+r,θ2=θ1-(h-η)(h-r).

By substituting k2=kT2 into T(k2), we can get a sufficient condition for the Turing instability as follows:

T(k2)=(D1(δ-h+2η-η2h+r)+D2(δ-1+η2h+θ2θ1(2+ηh(b-1))))k2+θ2θ1(2(δ-h)+(5-b)η+2r+(b-1)ηh(δ+3η+r))+(δ-1)(δ+2η+r-h)+η2h(1-b+r)+D1D2k4<0.

Remark 3.2

We can find out the critical value of bifurcation parameter δ for Turing bifurcation by replacing the inequality in (3.10) with the equality. Thus the critical value δT has to be satisfied with the following equation;

14D1D2(D1(δT-h+2η-η2h+r)+D2(δT-1+η2h+θ2θ1(2+ηh(b-1))))2=(δT-1)(δT+2η+r-h)+η2h(1-b+r)+θ2θ1(2(δT-h)+(5-b)η+2r+(b-1)ηh(δT+3η+r)).

At the Turing threshold δT, the spatial symmetry of the system is broken and the patterns are stationary in time and oscillatory in space with the wavelength

λT=2πkT.

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 k = kw ≠ 0 in equation (3.4). Thus we can get the conditions for the wave bifurcation in the following theorem.

Theorem 3.3

Wave bifurcation occurs if δ > δw, where

δw=L+hM2(b-1)(b+h-r-1).

Here,

L=h(D1+D2)(b-1)k2+2(b2(h-r)+(h-r-1)r+b(h2+r(r+2)-h(2r+1))),M=((D1+D2)(b-1))2k4+4b(h-r)(b+h-r-1).
Proof

In order that system (1.5) has a wave bifurcation, equation (3.4) must have purely imaginary roots when k ≠ 0. In other words, the wave bifurcation occurs if the following conditions are satisfied:

Im(λk)0and Re(λk)=0at k=kw0.

Thus, from elementary calculation, the critical value of wave bifurcation parameter δ can be obtained as

δw=L+hM2(b-1)(b+h-r-1)

where

L=h(D1+D2)(b-1)kw2+2(b2(h-r)+(h-r-1)r+b(h2+r(r+2)-h(2r+1))),M=((D1+D2)(b-1))2kw4+4b(h-r)(b+h-r-1).

Remark 3.4

In fact, at the wave threshold δ2, both spatial and temporal symmetries are broken and the patterns are oscillatory in space and time with the wave length λw satisfying

λw=2πkw,kw2=1D1+D2(PS+QI)(tr(Jk)=0).

In this section, we display numerical simulations of the spatiotemporal parasite-host system (1.5) to investigate spatiotemporal pattern formations caused by Turing bifurcation.

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 Δx = Δy = 0.25 and the time step Δt = 0.01 which satisfy the CFL(Courant - Friedrichs-Lewy) stability criterion for two dimensional diffusion equations [9, 13]. It is well known that spatiotemporal dynamics of a diffusion-reaction system depends on the choice of initial conditions [17]. It seems to be reasonable from the biological point of view that the initial density distribution is taken as a small amplitude random perturbation around the steady state E* = (S*, I*). We run the numerical simulations until the numerical solutions either reach a stationary state or show a behavior that does not seem to change its characteristic anymore. In the numerical simulation, we have figured out that the distributions of uninfected and infected population are always of the same type. Thus we pay attention to the pattern formation of the uninfected population.

In order to illustrate Turing patterns numerically, first, we fix parameter values in system (1.5) as follows;

b=0.1,h=1.1,r=0.1,D1=0.1,D2=1.0.

If we take the parameter value δ = 0.5, we can observe that the random initial distribution is transformed into a regular spotted pattern, as shown in Figure 1. On the other hand, in Figure 2, we can see the existence of the steady state having the spotted pattern and the stripelike pattern simultaneously when the value δ = 0.7 is chosen. Also Figure 3 shows that stripelike spatial patterns are prevalent in the whole domain eventually for the value δ = 0.9.

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 (1.5) accordingly to the value δ. In this study, we show that typical Turing patterns, which are well known in a predator-prey system [7, 18, 25], can be investigated in a parasite-host system as well.

Fig. 1. Snapshots of contour pictures of the time evolution of the uninfected population in system () when δ = 0.5: (a) 0 iteration; (b) 30000 iterations; (c) 49000 iterations
Fig. 2. Snapshots of contour pictures of the time evolution of the uninfected population in system () when δ = 0.7: (a) 0 iteration; (b) 15000 iterations; (c) 49000 iterations
Fig. 3. Snapshots of contour pictures of the time evolution of the uninfected population in system () when δ = 0.9: (a) 0 iteration; (b) 15000 iterations; (c) 49000 iterations
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