Global Analysis of a Simple Parasite-host Model with Homoclinic Orbits

In this paper, a simple parasite-host model proposed by Ebert et al.(2000) is reconsidered. The basic epidemiological reproduction number of parasite infection (R 0) and the basic demographic reproduction number of infected hosts (R 1) are given. The global dynamics of the model is completely investigated, and the existence of heteroclinic and homoclinic orbits is theoretically proved, which implies that the outbreak of parasite infection may happen. The thresholds determining the host extinction in the presence of parasite infection and variation in the equilibrium level of the infected hosts with R 0 are found. The effects of R 0 and R 1 on dynamics of the model are considered and we show that the equilibrium level of the infected host may not be monotone with respect to R 0. In particular, it is found that full loss of fecundity of infected hosts may lead to appearance of the singular case. 1. Introduction. Parasites can reduce host density and induce host population extinction in some cases. In order to understand how six microparasites regulate Daphnia populations and drive the populations to extinction, Ebert et al.[1] formulated the following epidemiological microparasite model

1. Introduction.Parasites can reduce host density and induce host population extinction in some cases.In order to understand how six microparasites regulate Daphnia populations and drive the populations to extinction, Ebert et al. [1] formulated the following epidemiological microparasite model where x(t) and y(t) represent the densities of uninfected (susceptible) and infected (infective) hosts at time t, respectively; a is the maximum per capita birth rate of uninfected hosts; θ(0 ≤ θ ≤ 1) is the relative fecundity of an infected host; K is the carrying capacity of the environment for the host population; d is the parasiteindependent host background mortality; β is the constant infection rate, and α is the parasite-induced excess death rate.
In model (1), the microparasite transmission is assumed via a mass action process, the fecundity of uninfected host is density-dependent, and the fecundity of infected host may be reduced due to being infected compared with that of uninfected host.
Here, θ = 0 means that infected hosts completely lose fecundity; θ = 1 means that the fecundity of infected hosts is not affected by parasite infection; 0 < θ < 1 means that the fecundity of infected hosts is reduced, but they still have certain fecundity.For model (1), there is always equilibrium O(0, 0), which is a saddle when a > d.This implies that extinction of host is impossible when a > d, that is, host always persists when a > d.But, in [1], the simulation for the stochastic model indicates that extinction of host likely occurs in certain parameter regions.
By carefully examining the infection term βxy, Hwang and Kuang [2] replaced the mass action incidence function βxy with a standard incidence function βxy/(x + y), and obtained the following model where β represents the maximum number of infections that an infected host can cause per unit time.Subsequently, based on model (2), some higher dimensional epidemiological models were presented and analyzed [3,4,5].Since model (2) is not differentiable at the origin, Hwang and Kuang [2] initially transformed model (2) into the form of a Gause-type predator-prey system by making the change of variables (x, y) → (u, y) with u = x/y, and then, by investigating the transformed system, they showed that the transformed system can exhibit the parasite induced host extinction.This theoretically complements the findings in [1].However, Hwang and Kuang [2] did not consider dynamical behaviors of model (2) directly, some complicated and interesting dynamics of model (2) were missed.
In [6], Berezovsky et al. incorporated the emigration of uninfected hosts into model (2), and then obtained the following model where m is the per capita emigration rate of uninfected hosts.They mathematically studied it as an epidemiological model, and found a family of homoclinic orbits by investigating the local dynamics of the model near the origin.But the effect of the related parameters on dynamical behaviors of the model is not discussed completely.
In particular, the case θ = 0 is not considered.Note that model (2) and model (3) are dynamically equivalent.
In this paper, we reconsider model (2) by rescaling it and completely analyzing the global dynamics of the rescaled model.The initial objective is to fully understand the asymptotic behavior of model (2) and theoretically prove the existence of its heteroclinic and homoclinic orbits.In particular, we investigate the effect of the basic epidemiological reproduction number of parasite (R 0 ) and the basic demographic reproduction number of infected hosts (R 1 ) on the survival of hosts, and analyze the change of the level of infected hosts with the basic reproduction number of parasite infection (R 0 ).
The organization of this paper is as follows.In the next section, we initially rescale model (2) and then introduce the primary results on the rescaled system.In Sections 3 and 4, we theoretically analyze the rescaled system with cases 0 < θ ≤ 1 and θ = 0, respectively, and prove the existence of heteroclinic and homoclinic orbits of the rescaled system.Dynamic behaviors of system (2) are demonstrated in Section 5, where the effect of R 0 and R 1 on dynamics of the model is investigated.The effect of R 0 on the level of infected hosts is also considered in Section 6.We conclude with a discussion of the results in Section 7. where It follows from (4) that for x ≥ 0 and y ≥ 0, which gives that lim t→+∞ (x(t), y(t)) = (0, 0) for δ ≥ 1.So, in the rest of this paper, we shall assume δ < 1, which implies that a > d for (2).For convenience of discussion hereafter, we give the following results about (4).
On the other hand, if we choose B(x, y) = 1/(xy) as a Dulac function, it follows from (4) that Therefore, we have Lemma 2.2.In the interior of the set D 1 , there is no closed orbit to (4).
Here, E 0 represents that host has not been infected, E * implies that host is infected chronically.About the local stability of E 0 and E * , we have Theorem 3.1.The boundary equilibrium E 0 is locally asymptotically stable on the set D 1 if s ≤ δ + r, and unstable if s > δ + r.The positive equilibrium E * is locally asymptotically stable if s > δ + r and θ(s − δ − r) > (δ + r)(s − r − 1).
Proof.From the Jacobian matrices of ( 4) at E 0 and E * , it is easy to know that E 0 is locally asymptotically stable on the set D 1 for s < δ + r, and unstable for s > δ + r.And the positive equilibrium E * is locally asymptotically stable if it is feasible.When s = δ + r, equilibrium E 0 is a higher order one.To discuss its locally stability on the set D 1 , we make the change of variables: u = x − (1 − δ), v = y, which moves E 0 in the xy-plane to the origin in the uv-plane, then (4) becomes It follows from dw dt = 0 that Direct calculation shows that, for the implicit function w = w(v) defined by (8), dw dv | (0,0) = 0, which implies w = o(v).Substituting it into the second equation of (7) gives dv/dt = −sv 2 /(1 − δ) + o(v 2 ).Thus, according to Theorem 7.1 in [7], s/(1 − δ) > 0 implies that E 0 is locally asymptotically stable on the set D 1 for s = δ + r.
To consider the global stability of E 0 and E * of (4) on the positively invariant set D 1 , we first need to discuss the dynamical behaviors of (4) near the origin in the first quadrant since it may be complex.Lemma 3.2.(i) When s ≤ r + 1, orbits of system (4) starting from near the origin in the interior of D 1 move away from the origin as t increases; (ii) When s > r + 1 and θ(s − δ − r) > (δ + r)(s − r − 1), orbits of system (4) initiating from near the origin in the interior of D 2 move away from the origin as t increases.
Proof.Let dt = (x + y)dτ then (4) becomes Using the polar coordinates x = ρ cos ϕ, y = ρ sin ϕ, (9) becomes where So system (9) has no characteristic direction in the first quadrant, which implies that all orbits of (4) starting from near the origin move in the clockwise direction.Since dx/dτ | y=0 > 0 for 0 < x < 1 − δ, then, according to the dependence of continuity of solutions on the initial conditions, all the orbits of (4) starting from near the origin in the interior of D 1 move away from the origin as t increases.
(ii) When s > r + 1 and θ(s − δ − r) > (δ + r)(s − r − 1), for (x, y) sufficiently close to the origin, we have and Thus, according to dependence relationship of solutions of (4) on the initial values, orbits of (4) initiating from near the line θy = (s − r − 1)x sufficiently close to the origin move away from the origin.And, for points in the interior of D 2 , the inequality (s − r − 1) cos ϕ < θ sin ϕ always holds, which implies G(ϕ) < 0. Similar to the case s ≤ r + 1, (9) has no characteristic direction in the interior of D 2 for this case.Therefore, when s > r + 1 and θ(s − δ − r) > (δ + r)(s − r − 1), orbits of system (4) starting from near the origin in the interior of D 2 move away from the origin as t increases.
Notice that when δ + r < s ≤ r + 1, the inequality θ(s − δ − r) > (δ + r)(s − r − 1) holds.So equilibrium E * exists when δ + r < s ≤ r + 1.Therefore, according to Lemma 2.2, we have the following conclusion about the global stability of E 0 and E * .Theorem 3.3.For system (4), when s ≤ δ + r, equilibrium E 0 is globally stable on the set D 1 ; when s > δ + r and θ(s , equilibrium E * is globally stable in the interior of the set D 1 . 3.2.Heteroclinic and homoclinic orbits.According to the results obtained in Section 3.1, the global dynamical behaviors of (4) have been understood clearly for the following cases: Notice that the inequality θ(s − δ − r) > (δ + r)(s − r − 1) always holds for δ + r < s ≤ r + 1, then, in the following, we consider dynamics of (4) in a case of s > 1 + r and θ(s We initially discuss the vertical isocline L of (4).Denote then, it follows from dx/dt = 0 that the vertical isocline L of ( 4) is defined by equation Notice that the discriminant of equation g(1, u) = 0 of u is also ∆.On the other hand, from θ(s Therefore, for this case, g(x, y) can be rewritten as g(x, y) = θ(y Furthermore, (12) can be rewritten as By applying Lemma 7.1 in Appendix to (14), we have the following Lemma with respect to the vertical isocline L of system (4).Lemma 3.4.When s > r + 1 and θ(s − δ − r) ≤ (δ + r)(s − r − 1), the vertical isocline L of (4) defined by (12) (or ( 14)) consists of two branches, which have the following properties: (i) The two branches intersect with two coordinate axes at points (1 − δ, 0) and (0, 1), respectively.
(ii) The two branches are located in the regions D 3 = (x, y) ∈ R 2 + : y < k − x and D 4 = (x, y) ∈ R 2 + : y > k + x , respectively.(iii) One of the two branches in the region D 3 is concave down, while the other in the region D 4 is concave left.
According to Lemma 3.4, the two branches of the vertical isocline L of (4) are shown in Figure 1.The heteroclinic and homoclinic orbits of (4).The thick closed curve is a heteroclinic closed orbit, the thin curves surrounded by the closed curve are all homoclinic orbits.Here, θ = 0.2, δ = 0.4, r = 0.5, and s = 1.8.All the orbits of (4) starting from the first quadrant approach the origin along the direction ϕ = arctan 1.5 as t tends to positive infinity.Theorem 3.5.When s > r + 1 and θ(s − δ − r) ≤ (δ + r)(s − r − 1), for (4) we have the following results: (i) There is a heteroclinic closed orbit of (4) on the set D 1 ; (ii) There is a family of homoclinic orbits of (4), which are full of the domain surrounded by the heteroclinic closed orbit(Fig.2); (iii) All the orbits of (4) starting from the first quadrant approach the origin along the direction ϕ = arctan(s − r − 1)/θ as t tends to positive infinity.
Proof.(i) We initially prove that there is a heteroclinic orbit L 1 of (4) in the interior of D 1 , where lim t→+∞ (x(t), y(t)) = (0, 0) and lim t→−∞ (x(t), Since s > r + 1 implies s > δ + r, it follows from the Jacobian matrix of ( 4) at E 0 that E 0 is a saddle.Hence, there is a saddle separatrix L 1 of E 0 in the first quadrant, whose slope at E 0 is It is easy to know that L 1 is an unstable manifold of E 0 , that is, lim t→−∞ (x(t), y(t)) = E 0 for (x(t), y(t)) ∈ L 1 .By Lemma 3.4, the vertical isocline L of (4) passes through E 0 .And direct calculation shows that the tangential slope of L at E 0 is Thus, it follows from (15) and ( 16) that −1 < k 1 < k 2 < 0, which implies that the separatrix L 1 near E 0 is located between the line x + y = 1 − δ and the vertical isocline L.
According to Lemma 2.1, when s > r+1, the separatrix L 1 must pass through the line y = (s − r − 1)x/θ and enter the region D 2 in a finite time.Again, s > r + 1 and θ(s so it follows from Lemma 3.4 that system (4) has no equilibrium in the set D 2 \{O}, then L 1 must approach the origin as t tends to positive infinity, that is, lim t→+∞ (x(t), The above inference shows that L 1 ⊂ D 1 is a heteroclinic orbit of (4) connecting equilibrium E 0 and the origin O.
(ii) Since (4) has no positive equilibrium in the set D 1 and ϕ = 0 is a characteristic direction of (4) according to expression (11), it is easy to know that there is a family of homoclinic orbits of (4) full of the interior of the heteroclinic closed orbit.
In fact, it follows from (17) that So any trajectory of system (17) starting from the set R 2 + \ D will enter the set D in a finite time, where Then it follows that lim sup t→+∞ u(t) ≤ (s − δ − r)/θδ.Therefore, the claim holds.
Again, system (17) has three nonnegative equilibria: Ō(0, 0), Ē0 (1 − δ, 0), and Ē1 (0, (s−r−1)/θ) for this case.They are all on the boundary of the region D. From the Jacobian matrices of (17) at equilibria Ō, Ē0 and Ē1 , it is easy to get that Ō is an unstable node, Ē0 is a saddle, and Ē1 is a stable node.Since (17) has no equilibrium in the interior of the set D, the equilibrium Ē1 is globally stable on the set D for this case.Therefore, for (4), lim t→+∞ x(t) = 0 and lim t→+∞ (y(t)/x(t)) = (s − r − 1)/θ.It implies that all the orbits of (4) starting from the first quadrant approach the origin along the direction ϕ = arctan(s − r − 1)/θ as t tends to positive infinity.4. Mathematical analysis for system (4) with θ = 0.In this section, we consider dynamical behaviors of the rescaled system (4) in the case θ = 0, which implies that the fecundity of infected hosts is completely lost.
When θ = 0, system (4) becomes Similar to the previous argument for the case of 0 < θ ≤ 1, for system (18) the first two items of Lemma 2.1 hold true, and so is Lemma 2.2.Wit respect to the existence and stability of equilibria of system (18), we have the following results.
Theorem 4.1.System (18) always has the infection-free equilibrium E 0 (1 − δ, 0), which is globally stable on the set D 1 when s ≤ δ + r; when δ + r < s < r + 1, system (18) has a unique positive equilibrium E * 1 (x * 1 , y * 1 ), which is globally stable in the interior of the set D 1 , where In the following, we consider dynamics of system (18) for the case s ≥ r + 1.When s ≥ r +1, it is easy to know that, in the first quadrant, the vertical isocline of (18) defined by 1−δ = (x+y)+sy/(x + y) is above its horizontal isocline defined by sx = (δ + r)(x + y), and they intersect only at the origin.Especially for the case s = r + 1, they are tangent at the origin.So system (18) has no positive equilibrium as s ≥ r + 1.And equilibrium E 0 is a saddle as s ≥ r + 1.
Notice that both x-axis and y-axis are solution curves of system (18).For system (18) we can have the following results similar to Theorem 3.5.Theorem 4.2.When s ≥ r + 1, system (18) has a heteroclinic closed orbit on the set D 1 , and a family of homoclinic orbits is full of the interior the heteroclinic closed orbit(Figure 3).
For system (18), function G(ϕ) defined in (11) becomes G(ϕ) = ρ(s − r − 1) sin ϕ cos ϕ(cos ϕ + sin ϕ), then, when s > r + 1, ϕ = 0 and ϕ = π/2 are the characteristic directions of system (18); when s = r + 1, G(ϕ) ≡ 0, which is the singular case.Therefore, for dynamics of system (18) near the origin O we have Corollary 1.When s > r + 1, any orbit of system (18) starting from the first quadrant approaches the origin along the y-axis as t tends to positive infinity, and any orbit leaving the origin moves along the x-axis as t increases(Figure 3(a)); however, when s = r + 1, along any direction except for ϕ = arctan s−(δ+r) δ+r , there is a unique orbit of system (18) which either tends to or leaves the origin(Figure 3(b)) as t increases.Figure 3.The heteroclinic and homoclinic orbits of (18).The thick closed curve is a heteroclinic closed orbit, the thin closed curves surrounded within the heteroclinic closed orbit are all homoclinic orbits.Figure 3(a) corresponds to the case s > r + 1, where δ = 0.4, r = 0.5, and s = 1.8. Figure 3(b) corresponds to the case s = r + 1, where δ = 0.4, r = 0.5, and s = 1.5..

Dynamic behaviors of system (2)
. In sections 3 and 4, we have completely analyzed the rescaled system (4).In this section, we will discuss dynamical behaviors of model ( 2) according to the above results, and explain these results epidemiologically.
We first summarize the main results on system (4) in Table 1, which shows that system (4) has four types of dynamical behaviors: For Case (C1) defined in Table 1, the global stability of the origin O implies that hosts (including uninfected and infected hosts) go to extinction eventually.The host extinction in such case is not due to the parasite infection but demographic decline feature of hosts themselves.
For Case (C2), the global stability of E 0 implies that parasite infection dies out eventually in the host population, while uninfected host population approaches a positive constant as t tends to infinity.
For Case (C3), the parasite infection keeps present persistently.The sizes of both uninfected and infected hosts stabilize to positive constants eventually as t tends to infinity.
For Case (C4), the parasite infection leads to host extinction due to parasite regulation, whereas hosts do not extinct in the absence of parasite infection, which is different from Case (C1).Here, the existence of homoclinic orbits shows that the outbreak of parasite infection is possible.Whether the infected hosts can reproduce may lead to various phenomena.For θ = 0, that is, the infected hosts can certainly reproduce, by Theorem 3.5 we have lim t→∞ y(t)/x(t) = (s − r − 1)/θ, which implies that in the process of host extinction the ratio between sizes of infected and uninfected hosts tends to a constant.However, for θ = 0, that is, the infected hosts can not reproduce, the change of the ratio is complex according to Corollary 1.
To describe the obtained results with the original parameters in model ( 2) and demonstrate them epidemiologically, we introduce three new parameters with the original parameters: Since 1/d and 1/(d + α) are the average periods that the uninfected and infected hosts stay in their compartments, respectively.a and θa are the per capita maximum birth rates of uninfected and infected hosts, respectively.Then R 0 is referred to as the basic epidemiological reproduction number of parasite infection, that is, average of number of secondary infections induced by a single infected host in a whole susceptives during infectious period.Whereas, R 1 and R 2 are the basic demographic reproduction numbers of infected and uninfected hosts, respectively.For simplicity, we define the following two parameters where γ represents the ratio between the average life spans of infected and uninfected hosts, R0 is a threshold determining the existence of hosts in the presence of parasite, which will be understood in the following discussion.By using expressions (5) and the new parameters defined above, the inequality θ(s Obviously, (19) holds true for R 0 > 1 and R 2 > 1 when R 1 ≥ 1.However, when R 1 < 1, from (19) we have O is globally attractive, and there are a family of homoclinic orbits of system (4).Table 1.The complete global results of system (4).

Cases
Conditions O is globally attractive, and there are a family of homoclinic orbits of model ( 2).Table 2.The complete global results of model ( 2) O is globally attractive, and there are a family of homoclinic orbits of system (18).Table 3.The complete global results of system (18).then divided into two subcases, and accordingly, Especially, when θ = 0, that is, R 1 = 0, Table 2 can become Table 3, where Since R 2 ≤ 1 implies that the basic demographic reproduction number of uninfected hosts is not greater than 1, then, corresponding to Case (C1), the extinction of hosts is natural in the absence of parasite infection.Therefore, in the following we only discuss the case R 2 > 1, which implies that hosts can persist forever in the absence of parasite infection.
In order to understand the dependence of dynamics of model ( 2) on the parameters R 0 and R 1 for the given parameters R 2 (R 2 > 1) and γ (γ < 1), we partition the feasible regions of R 0 and R 1 into some subregions according to Table 2.These regions are shown in Figure 4. Figure 4(a) and (b) correspond to two cases: R 2 γ ≥ 1 and R 2 γ < 1, respectively.
In the region Ω 2 (or Ω 2 ), the basic reproduction number of parasite infection R 0 is less than unity, which implies the parasite infection dies out eventually in the host population.The uninfected host population tends to K(a − d)/a as t tends to infinity.
In the regions Ω 31 , Ω 32 and Ω 4 (or Ω 32 and Ω 4 ), the basic reproduction number of parasite infection is greater than 1, which indicates that the parasite infection keeps persistently.However, for the different values of R 1 , there are some essential differences between the associated dynamical behaviors of (2).
In the region Ω 31 , R 1 ≥ 1 implies that infected hosts can reproduce sufficient uninfected hosts to maintain the supply of uninfected hosts being infected.In such case, parasite infection can not regulate the host population even if R 0 is sufficiently large.Whereas, in Ω 32 (or Ω 32 ), R 1 < 1 implies that infected hosts' reproduction is not large enough to induce various dynamics, depending on relation of R 0 and R 1 .In such scenario, parasite infection can not regulate the host population unless R 0 ≥ R0 .
In the region Ω 4 (or Ω 4 ) where R 0 ≥ R0 and R 1 < 1, a combination of large epidemiological reproduction number and small demographic reproduction number of infected hosts results in ultimate extinction of both uninfected and infected hosts due to parasite regulation.However, according to Theorem 3.5 and Corollary 1, the trend of the extinction of uninfected and infected hosts is different for the cases which implies that the extinction of uninfected and infected hosts is along certain direction.Whereas, when R 1 = 0, the limit (i.e., the direction of extinction) depends on the initial values of (2).We initially consider the effect of R 0 on dynamics of the system for various R 1 .When R 1 < 1, the dynamics of system (2) go through extinction of parasite infection (i.e., global stability of disease-free equilibrium for R 0 ≤ 1), persistence of both parasite infection and host population (i.e., global stability of the endemic state for 1 < R 0 < R0 ), and hosts extinction while infection persistence (i.e., global attractiveness of the origin for R 0 ≥ R0 ) as the epidemiological reproduction number R 0 increases.Whilst R 1 ≥ 1, the first two kinds of dynamics of system (2) certainly occur, but the third kind -parasite regulation does not happen.For fixed R 0 (either R 0 < 1 or 1 < R 0 < 1 + γ(R 2 − 1)), global extinction or persistence of parasite infection is independent on R 1 .However, for relatively large R 0 (R 0 ≥ 1+γ(R 2 −1)), increasing R 1 would change the dynamics of system (2) from host extinction to host persistence while keeping infection present.

6.
The effect of R 0 on the equilibrium level of infected hosts.For classical parasite-host models, virus dynamical models and epidemic models, the level of infected hosts(virus, individuals) often increases with increase of the associated basic reproduction number when the positive equilibrium is feasible.But, for system (2) the dependence of the level of infected hosts on the associated basic reproduction number is not such simple.In the following we consider the change of the equilibrium level of infected hosts with the basic reproduction number of parasite infection(R 0 ).
On the other hand, Since the existence of the positive equilibrium E * corresponds to cases (C3 1 ) and (C3 2 ) in Table 2, then, for R 0 > 1 and R 2 > 1, we list a table (Table 4) to classify the feasible regions of parameters and the associated effect of R 0 on the level of infected hosts which are shown in Figure 5. Figure 5(a) shows the fact that y * increases for 1 < R 0 < R 0 and decreases for R 0 < R 0 < R0 < +∞, and lim R0→ R0 y * = 0; Figure 5(b) is similar to Figure 5(a), but the limit of y * as R 0 → R0 is K(R 1 − 1)/R 1 ; Figure 5(c) represents that y * increases monotonously, and lim R0→∞ y * = K(R 1 − 1)/R 1 .
Comparing Figures 5 (a), (b) and (c) suggests the common feature is that the equilibrium level of the infected host (y * ) increases monotonously as R 0 increases initially.This can be easily understood epidemiologically.However, as R 0 keeps increasing various trends of y * can observed, depending on R 1 .In fact, Figures 5  (a), corresponding to R 1 < 1, shows the more R 0 and the less the equilibrium level of the infected hosts.That is because the size of uninfected hosts reproduced by infected ones is small, which leads to the shortage of uninfected hosts.Then large R 0 causes more infected hosts and consequently more disease-induced death, and hence host extinction due to parasite infection may happen.For large R 1 (R 1 ≥ R1 ), the infected hosts can reproduce sufficient uninfected ones to not only maintain supply of resource being infected but also balance disease-induced death, which suggests that y * persistently increases and stabilizes at a fixed level with increasing R 0 , as shown in Figures 5 (c).For middle values of R 1 (1 ≤ R 1 ≤ min R 2 γ, R1 ), the equilibrium level of the infected hosts initially reaches a maximum and then decline to a certain level as R 0 increases.In addition, it follows from Figure 5 (a) and (b) that R0 is a threshold in the sense of monotonicity of y * .7. Conclusion and discussion.In this paper, by rescaling parasite-host model (2), we mathematically analyzed the global dynamics on the feasible region, and theoretically proved the existence of heteroclinic and homoclinic orbits which implies that the infection may break out in the process of host extinction.
According to the obtained results on model (2), we demonstrated the effect of the basic epidemiological reproduction number (R 0 ) and the demographic reproduction number of infected hosts (R 1 ) on the dynamics of the model and the level of infected hosts.Here, we obtained two new findings: one is that a combination of R 0 and R 1 may cause the complexity of dynamics of the model, the other is that, for the different range of R 1 , variation in the equilibrium level of the infected host (y * ) with R 0 may not be monotone, which is different from the classical models accepted by mathematicians and epidemiologists.In addition, when the fecundity of infected hosts is lost fully (i.e., θ = 0), the appearance of the singular case implies that the trends of host extinction may depend on the initial state of the model.Note that in our model the standard incidence function βxy/(x + y) plays an important role in coming up with rich dynamics from the point view of mathematics.In fact, standard incidence is indeed based on the epidemiological meanings, where finite contacts of an individual making in a unit time are observed due to finite and often slow movement in large populations, and more details can be found in [2].We model the growth of host population with logistic growth function to describe the density-constraint growth within host population (or intra-species competition for resources).Hence, our model is reasonable to describe the realities in both biology and epidemiology.Further, our main results show that parasite infection could die out or persist in certain conditions, which is similar to those for other simple models [8].However, it is interesting to note that our model (2) exhibits some novelties because of introduction of standard incidence.In particular, host extinction can be induced by either demographic decline feature of hosts themselves or parasite regulations.During host going to extinction parasite infection may outbreak and persist eventually, which is different from that for the models with bilinear incidence.Hence, the model examined here describes biological/epidemiological phenomena more reasonably and our conclusions show the dynamics and the biological implications more extensively.(20) consists of two branches, which are in the regions D1 = (x, y) ∈ R 2 + : y < nx and D2 = (x, y) ∈ R 2 + : y > mx}, respectively.Except for the origin, the two branches intersect with two coordinate axes at points (mn/p, 0) and (0, 1), respectively.One Notice that v < u/m is equivalent to y > mx, then the region D 2 on the (u, )vplane corresponds to the region D2 on the (x, y)-plane.Thus, the part of the curve L in the region D2 is concave leftwards.
Figure2.The heteroclinic and homoclinic orbits of (4).The thick closed curve is a heteroclinic closed orbit, the thin curves surrounded by the closed curve are all homoclinic orbits.Here, θ = 0.2, δ = 0.4, r = 0.5, and s = 1.8.All the orbits of (4) starting from the first quadrant approach the origin along the direction ϕ = arctan 1.5 as t tends to positive infinity.

1 .
For m > n > 0 and p > 0, the part of curve L in the first quadrant defined by equation(y − mx)(y − nx) = (y + px)(y + x) 2 Table 1 can be re-expressed with Table 2, in which Case (C3) in Table 1 is