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provided r + μ − σ > 0, an assumption that we are making from now on. Substituting into the second equation (10) we find

The right hand side is positive if (σ − μ) f + μg > 0 and from this the above restriction can

Ecoepidemics with group defense

k=0

where b3 = 1,

b2 = σ (q − w) K (g − f)2 , b1 = 2f (g − f) σ (q − w) K (wσ − rw − qμ) K (g − f)2 , b1 = f2σ (q − w) K + 2f (g − f) (wσ − rw − qμ) K, b0 = f2 (wσ − rw − qμ) K + m2rw.

Elena Cagliero, Ezio Venturino

It has always a real root, and we seek now su cient conditions for a nonnegative real root. Since q w, it follows that

lim P (A) = −∞,

A→∞

so that if the constant term is positive, at least one positive real root must exist. This occurs if

In summary the equilibrium E5 = (A5, T5, U5) arises with first component given by the positive root of (17) and the remaining ones by (12) and (15), which need to be nonnegative, and further feasibility conditions given by (19) or (20).

5Stability

The elements of the Jacobian matrix J = (Jik), i, k = 1, 2, 3 are

Observe that since A = S(S + I)−1 1 and q < w two of the above terms have a fixed sign:

Ecoepidemics with group defense

having used the fact that (σ − μ) f + gμ > 0 and the first condition (7). The other two eigenvalues are the roots of

In view of the feasibility conditions (7), the Routh-Hurwitz stability conditions for (23) hold. Stability of E2 is therefore regulated only by (22).

It is negative if and only if g2K [q (σ − μ) − rw] < −m2rw. But this cannot happen if q (σ − μ) − rw 0. Conversely, we are lead to the stability conditions

From the (strict) feasibility conditions (9) for E3, the constant term is always positive. Imposing that also the coe cient of the linear term is positive, we obtain the second stability condition,

Elena Cagliero, Ezio Venturino

For the equilibrium E4 some of the Jacobian entries, in view of the feasibility conditions (14) have fixed signs, as follows

while the remaining two must agree, since the same factor appears in the two elements, although the sign is not decided:

We now study the signs of J431 and J433 . Considering J431 and using the feasibility condition (14) we find that for its positivity we must have

But this contradicts the feasibility condition (14), so that it must be negative. In summary we then have

Ecoepidemics with group defense

The characteristic equation is now a cubic,

3

akλk ≡ λ3 − (C + G) λ2 − (ZB + F D − CG) λ − Z (ED − BG) = 0. (28)

k=0

Using the signs of Z, B, C, D, E, F and G all the coe cients ak, k = 0, . . . , 3 are positive. We can thus use the Li´enard-Chipart criterion, a particular case of the Routh-Hurwitz criterion, thereby determining the sign of the eigenvalues imposing that the following de-

We can conclude for this case that E4 is stable if (29) holds.

Stability of E5 is investigated numerically.

6Bifurcations

Note that transcritical bifurcations further arise between E1, E2 and E3.

We then try to establish if there are special parameter combinations for which Hopf bifurcations arise. For this purpose, we need purely imaginary eigenvalues. This is easy to assess for a quadratic characteristic equation, λ2 + bλ + c = 0 since we need the linear term to vanish, b = 0, and the constant term to be negative, c < 0. For a generic cubic of the

but these conditions contradict each other. We conclude that at E2 no Hopf bifurcations can carise.

At E3 the characteristic equation factors, and the quadratic (25) from feasibility (9) has a positive constant term. Imposing that the linear term vanishes, we find the value

m 2

g

(32)

this situation and investigate the remaining

Elena Cagliero, Ezio Venturino

7Equilibria interpretation

At equilibrium E1, we have U1 = 0 so that P1 = 0 and the predators vanish. Further,

A1 = 1 implying that I1 = 0. Thus only healthy prey survive, at the environment’s carrying

√

capacity, due to the model assumption of logistic growth, T1 = K indeed implies in this case S1 = K.

At E2 the request that A < 1 tells us that neither healthy nor infected prey disappear from the system, while, as in the previous case, all the predators die since U2 = 0. Therefore the disease remains endemic among the prey, while predators do not survive. Note once again that the the point E2 becomes equilibrium E1 if we assume that the disease transmission rate equals the disease mortality rate. In such case thus the disease can be eradicated.

At E3 we have again that A3 = 1, so that I = 0 and in this case the disease gets eradicated from the ecosystem, while the predators and healthy prey survive together. This is the only equilibrium for which we have proved analytically the existence of bifurcations, for the particular value of the prey carrying capacity K† = 3m2g−2.

At E4 and E5 we have coexistence, with the point E3 being a particular case of the latter equilibria, when A = 1. Further E4 and E5 di er because in the first case q = w, i.e. the infected and healthy prey are hunted at the same rate by predators, and therefore it can be regarded as a special case of E5. As for the latter, note that for the particular situation in which f2K (rw + qμ − wσ) = m2rw we find A5, as the cubic (17) goes through the origin. This implies that the healthy prey are wiped out. Therefore in this situation the ecosystem thrives, with predators and only infected prey.

8Simulations

The equilibrium E1 represents the situation where the only population which survives in the habitat is represented by the healthy prey. The fact that infected individuals are extinguished is consistent with the stability conditions of E1. In fact, the latter require that the disease incidence be lower than the disease-related mortality rate. Thus infected individuals die faster than they are recruited and ultimately there are not enough infectious individuals to propagate the disease. Its stable behavior is shown in Figure 1 for the parameter values σ = 0.2, r = 0.5, K = 5, μ = 0.4, q = 0.2, w = 0.5, m = 0.8, g = 0.1, f = 0.3.

At the equilibrium E2 predators get extinguished, but the disease remains endemic. In this situation the opposite condition of equilibrium E1 must be verified, namely the disease-related mortality rate is lower than the disease incidence. This suggests that it is reasonable to expect that the population of infected prey survives. Figure 2 contains a simulation leading to this equilibrium for the parameter values σ = 0.5, r = 0.5, K = 5,

Ecoepidemics with group defense

First model: first equilibrium

Figure 1: The stable equilibrium E1 is achieved for the parameter values σ = 0.2, r = 0.5, K = 5, μ = 0.4, q = 0.2, w = 0.5, m = 0.8, g = 0.1, f = 0.3.

μ = 0.4, q = 0.2, w = 0.5, m = 0.8, g = 0.1, f = 0.3..

The third equilibrium E3 shows coexistence of predators and healthy prey, with the disease eradicated. Figure 3 shows it for the parameter values σ = 0.5, r = 0.5, K = 5, μ = .4, q = 0.2, w = 0.5, m = 0.2, g = 0.1, f = 0.3..

The equilibrium E4 represents the possibility that all the populations in the system survive, i.e. E4 represents the coexistence equilibrium. The simulation of Figure 4 shows it for the following parameter values: σ = 0.4, r = 0.5, μ = 0.2, q = 0.5, w = 0.5, m = 0.3, f = 0.2, g = 0.1, K = 10..

In the study of the stability, focusing on the bifurcations, we discovered that, around E3, limit cycles should arise when K crosses the threshold value K† = 3m2g−2. In Figure 5 we present a simulation of the two-dimensional limit cycle for the parameter values σ = 0.5, r = 0.5, μ = 0.4, q = 0.2, w = 0.5, m = 0.2, g = 0.1, f = 0.3. Oscillations appear

Elena Cagliero, Ezio Venturino

First model: second equilibrium

Figure 2: Equilibrium E2 is obtained for the parameters: σ = 0.5, r = 0.5, K = 5, μ = 0.4, q = 0.2, w = 0.5, m = 0.8, g = 0.1, f = 0.3..

only for the second and the third variables, while for first one remains at the fixed level A = 1, to mean that the system is disease-free. Predators survive together with the healthy individuals but with persistent oscillations of the two populations. In Figure 6 a threedimensional phase-space portrait of the limit cycle is given.

For the equilibrium E5 our extensive simulations seem to indicate its instability.

9Conclusions

In this paper we studied an ecoepidemic model in which two popolations interact: the prey and the predators. We assumed that among the prey a disease develops, which spreads by contact. Further, the disease is unrecoverable, i.e. infected prey cannot heal from it.

Ecoepidemics with group defense

First model: third equilibrium

Figure 3: The equilibrium E3 is here shown for the parameter values σ = 0.5, r = 0.5, K = 5, μ = .4, q = 0.2, w = 0.5, m = 0.2, g = 0.1, f = 0.3..

Predation a ects for healthy and infected prey. The prey group together in a herd and exert some defensive strategy, for which mainly the individuals on the boundary of the herd su er from the attacks of the hunting population.

The original system formulation leads to possible singularities in the Jacobian, when the prey population vanishes. Therefore we have performed several changes of dependent variables to obtain a singularity-free reformulation. The newly obtained variables represent respectively the ratio of healthy prey over the total amount of prey, the number of predators per prey staying at the edge of the herd area and finally the number of prey occupying the edge of the herd.

We have discovered that there are four possibly stable equilibria, at which only the healthy prey thrive, or the disease remains endemic with only the prey population surviving, or healthy prey coexist, possibly with persistent oscillations, with the predators. The final