ISSN: 2394-7772 International Journal of Biomathematics and Systems Biology Official Journal of Biomathematical Society of India Volume 1, No. 2, Year 2015 A two-prey – diseased predator ecosystem ⋆ Barbara Bonaf` e, Alice Conchin Gubernati, Giulia Ricci, Ezio Venturino Dipartimento di Matematica “Giuseppe Peano”, Universit` a di Torino, via Carlo Alberto 10, 10123 Torino, Italy Abstract. In this paper we consider a disease-affected specialist predator that feeds on two re- sources. While the ecosystem is never wiped out, interestingly, just one prey cannot thrive in it. Transcritical bifurcations relate the simplest equilibria. Some sufficient conditions for the fea- sibility of the coexistence equilibrium are derived. In the particular case of the disease being unrecoverable, some additional healthy-predator-free equilibria are discovered, in which either one or both prey thrive, together with the infected predators. Key words: Ecoepidemiology, predator-prey, transmissible disease, two resources, stability. 1 Introduction Ecoepidemic models investigate the relationships between populations in which diseases play a substantial role, one first paper in this domain being [18]. An account for the developments of this research field is contained in [29] and the more recent [34]. In ecoepidemiology, which joins demographic models with epidemic ones, [19, 20, 24, 14, 16, 26], disease can be considered a way of controlling one, or both, interacting populations, [2], especially if one of them is considered a pest, [4, 1, 30, 27], or when one population is not really affected by a disease, but may cause harm to the other one which is considered a resource, [10, 11]. Many models have been formulated in the course of the years. An attempt for a comparative study has been performed in [2]. However, not just predator-prey ecosystems have been investigated, see for instance [31] for a case involving competing populations. Further, recently, systems have been investigated leading to more complicated behaviors, [3]. To this end, we also cite some attempts at looking at food webs, [5, 6, 7, 13]. Note however that in the literature also a kind of symmetric case has been considered, in which two diseases affect the ecosystem, [12, 25]. We consider here a predator-prey ecoepidemic model in which the predator is affected by a disease and can feed on two different types of prey. In the absence of either one of them, the model reduces to well-known models in the literature, see e.g. [33, 9]. One example of such a situation in real life is described for instance as follows. Red foxes, (Vulpus vulpus), are omnivores primarily feeding on small rodents, e.g. voles such as (Myodes glareolus), squirrels e.g. (Ratufa macroura, Glaucomys volans), [23] p. ⋆ The project was partially supported by the projects “Metodi numerici in teoria delle popolazioni” and “Metodi numerici nelle scienze applicate” of the Dipartimento di Matematica “Giuseppe Peano” of the Universit` a di Torino. 1 Corresponding Author E-mail: email: [email protected]Received on 11 Oct 2015 Accepted on 15 Dec 2015
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ISSN: 2394-7772International Journal of
Biomathematics and Systems BiologyOfficial Journal of Biomathematical Society of India
Volume 1, No. 2, Year 2015
A two-prey – diseased predator ecosystem ⋆
Barbara Bonafe, Alice Conchin Gubernati, Giulia Ricci, Ezio Venturino
Dipartimento di Matematica “Giuseppe Peano”, Universita di Torino,
via Carlo Alberto 10, 10123 Torino, Italy
Abstract. In this paper we consider a disease-affected specialist predator that feeds on two re-
sources. While the ecosystem is never wiped out, interestingly, just one prey cannot thrive in it.
Transcritical bifurcations relate the simplest equilibria. Some sufficient conditions for the fea-
sibility of the coexistence equilibrium are derived. In the particular case of the disease being
unrecoverable, some additional healthy-predator-free equilibria are discovered, in which either
one or both prey thrive, together with the infected predators.
Key words: Ecoepidemiology, predator-prey, transmissible disease, two resources, stability.
1 Introduction
Ecoepidemic models investigate the relationships between populations in which diseases play a substantial role, one first paper in
this domain being [18]. An account for the developments of this research field is contained in [29] and the more recent [34]. In
ecoepidemiology, which joins demographic models with epidemic ones, [19, 20, 24, 14, 16, 26], disease can be considered a way
of controlling one, or both, interacting populations, [2], especially if one of them is considered a pest, [4, 1, 30, 27], or when one
population is not really affected by a disease, but may cause harm to the other one which is considered a resource, [10, 11]. Many
models have been formulated in the course of the years. An attempt for a comparative study has been performed in [2]. However,
not just predator-prey ecosystems have been investigated, see for instance [31] for a case involving competing populations. Further,
recently, systems have been investigated leading to more complicated behaviors, [3]. To this end, we also cite some attempts at looking
at food webs, [5, 6, 7, 13]. Note however that in the literature also a kind of symmetric case has been considered, in which two diseases
affect the ecosystem, [12, 25].
We consider here a predator-prey ecoepidemic model in which the predator is affected by a disease and can feed on two different
types of prey. In the absence of either one of them, the model reduces to well-known models in the literature, see e.g. [33, 9].
One example of such a situation in real life is described for instance as follows. Red foxes, (Vulpus vulpus), are omnivores
primarily feeding on small rodents, e.g. voles such as (Myodes glareolus), squirrels e.g. (Ratufa macroura, Glaucomys volans), [23] p.
⋆ The project was partially supported by the projects “Metodi numerici in teoria delle popolazioni” and “Metodi
numerici nelle scienze applicate” of the Dipartimento di Matematica “Giuseppe Peano” of the Universita di Torino.1 Corresponding Author E-mail: email: [email protected]
Received on 11 Oct 2015Accepted on 15 Dec 2015
2 B. Bonafe, A. Conchin Gubernati, G. Ricci, E. Venturino
513-524. but they feed also on birds, mainly passeriformes and waterfowl, as well as raccoons, opossums, reptiles, [15] p. 529, or even
or small ungulates, [22]. Clearly these various prey populations have different habitats and do not interact directly for their search of
food, so that our demographic assumptions are satisfied.
On the other hand, foxes are affected by a number of diseases, the most famous one being rabies, but they have been found to be
affected by arthritis, [21] p. 421-422, leptospirosis and tularemia, and are also vectors for brucellosis and tick-born encephalitis. They
are even infected by Yersinia pestis, [23] p. 547. Also parasites are found in the fox guts, e.g. nematodes such as Toxocara canis and
Uncinaria stenocephala, Capillaria aerophila and Crenosoma vulpis, [28, 32].
Several other similar situations could be described in nature. For a panorama of various diseases affecting populations living
on the ground or in the aquatic medium, or even avian species, see [17]. In this paper however our aim is not to discuss a specific
ecosystem, but rather focus on the general properties of a system built on and containing these features.
The model is presented in the next Section, its equilibria are analyzed in Section 3 and some of their particular cases in the
Subsection 3.2. Section 4 contains the local stability analysis, together with the one of the particular cases. Then a Section for the
numerical simulations follows. A final discussion concludes the paper.
2 The model
Let R and U denote the two prey populations that live in the same environment of a predator population. Let F be the healthy predators,
while V denote those that are disease-affected. All the parameters are assumed to be nonnegative.
R′ = R
[a
(1−
R
K
)− cF − fV
], U ′ =U
[b
(1−
U
H
)−dF −gV
], (2.1)
F ′ = F [−m+ ecR+ edU −λV ]+νV, V ′ =V [λF + e f R+ egU − (ν+m+µ)] .
The first two equations describe the dynamics of the prey. We assume that these two populations do not interfere with each other, having
different habitats, although sharing the same physical location, as stated above. They reproduce logistically, with the environment
providing respective carrying capacities at levels K and H. These populations are subject to predators’ hunting: both healthy and
diseased predators hunt, at different rates, respectively c and f on the prey R and at respective rates d and g on the population U . The
third equation describes the healthy predators dynamics. We assume that the populations R and U are their sole source of food: in their
absence, the predators experience natural mortality at rate m. They convert captured food, from either one of the prey populations, into
newborns, with conversion factor 0 < e < 1, which is assumed to be the same for both healthy and infected predators. The “successful”
contact with an infectious individual moves them into the infected class, the disease contact rate being λ. They can also recover from
the disease, so that at rate ν the infected reappear among the susceptibles. The infected predators have a similar dynamics as far as
food intake is concerned, but they have a reversed behavior as far as entering their class: they do it at rate λ upon “successful” contact
among a susceptible and an infectious, and leave it at rate ν. But in addition, they also experience a disease-related mortality at rate µ.
The fact that hunted prey is transformed into diseased newborns follows from the assumption that we make, namely that the disease is
vertically transmitted.
3 The system’s equilibria
3.1 Case of the recoverable disease
The model (2.1) admits the following equilibria E(ν)i =(R
(ν)i ,U
(ν)i ,F
(ν)i ,V
(ν)i ), where the superscript emphasizes that these are obtained
when the disease is recoverable, ν 6= 0. These equilibria exist also in the particular case of no disease recovery ν = 0, some of them
will be explicitly obtained only in this situation and are discussed in the following Subsection 3.2, omitting in that case the superscript.
Easily, we find E(ν)0 = (0,0,0,0), E
(ν)1 = (0,H,0,0), E
(ν)2 = (K,0,0,0), E
(ν)3 = (K,H,0,0). These are always feasible. We then
have E(ν)4 =
(m(ec)−1,0,a(ecK −m)(Kec2)−1,0
)which is feasible for
ecK ≥ m. (3.1)
Then, the symmetric equilibrium E(ν)5 =
(0,m(ed)−1,b(edH −m)(Hed2)−1,0
), feasible for
International Journal of Biomathematics and Systems Biology 3
edH ≥ m. (3.2)
Next, E(ν)8 is found by solving for R and U as functions of F the first two equations of (2.1); substituting into the third one, we
then find:
R(ν)8 =
(a− cF8)K
a, U
(ν)8 =
(b−dF8)H
b, F
(ν)8 =
ab(Kec+Hed −m)
e(ad2H +bc2K).
Replacing the value of F(ν)8 thus found into the expressions for R
(ν)8 and U
(ν)8 we obtain the two explicit expressions, leading to
E(ν)8 =
(K(mcb− edHcb+aed2H)
e(ad2H +bc2K),
H(mda− ecKda+bec2K)
e(ad2H +bc2K),
ab(Kec+Hed −m)
e(ad2H +bc2K),0
)
which is feasible for the following nonempty conditions, as will be seen in Section 6:
e(Kc+Hd)> m, mda+bec2K > ecKda, mcb+aed2H > edHcb. (3.3)
The study of the equilibrium E(ν)10 with U
(ν)10 = 0 is performed as an intersection of suitable curves, as follows. In fact, this is the
coexistence equilibrium of the one-prey-only subsystem. In that respect, this model has been introduced long ago, [33]. But there the
analysis of coexistence is not decisive. Here instead we discuss the existence of the equilibrium in a different way, using a method
based on graphical tools that ultimately gives some sufficient conditions for its existence.
From the fourth equilibrium equation of (2.1), we find F as function of R,
F =A− e f R
λ, A = ν+m+µ (3.4)
Using this result in the first equilibrium equation, we obtain V as function of R:
V = jR+w ≡ce f K −aλ
K f λR+
aλ− cA
f λ. (3.5)
This is a straight line, intersecting the R axis at the point with abscissa Z = K(cA− aλ)(ce f K − aλ)−1. From this consideration and
(3.4) necessary conditions for the feasibility of the equilibrium are thus
R <A
e f; and either R > Z, for: ce f K > aλ, or R < Z, for: ce f K < aλ. (3.6)
Substituting F also into the third equilibrium equation we find the equation
−e2 f cR2 + e f λRV +(e f m+ ecA)R+λ(−m−µ)V −mA = 0 (3.7)
which represents a conic section.
For a generic conic section of the form pR2 + 2qRV + rV 2 + 2sR + 2tV + u = 0, the invariants are defined as C = pr − q2,
D = pru+2tsq− pt2 − rs2 −uq2. In our case they give
D =1
4
[e2 f λ2(ν−A)( f m+ cA)+ e2 f cλ2(ν−A)2 +mAe2 f 2λ2
]=
1
4νe2 f λ2( f m− cm− cµ), C =−
1
4e2 f 2λ2
< 0.
Since C < 0, excluding the degenerate case D = 0, the conic is a hyperbola. It intersects the R axis at the positive abscissae R1 =
m(ec)−1, R2 = A(e f )−1. For simplicity, since we are looking only for sufficient conditions, and do not aim at a complete study of all
the possible cases that can arise, we will assume that R2 > R1, i.e.
c(m+µ+ν)> f m. (3.8)
The intersection with the V axis occurs at the negative abscissa V1 =−mA[λ(m+µ)]−1. Its center is located at
(x0,y0) =
(tq− rs
pr−q2,
sq− pt
pr−q2
)=
(m+µ
e f,
c(m+µ−ν)− f m
f λ
).
There are several possible cases that can arise, for the various positions of the hyperbola and of the straight line (3.5). However, note
that x0 > 0. Three cases are possible, reported in Figures 1-3. In each picture, note that we show the three possible positions of the
straight line (3.5). We consider in particular Figure 1. In this case observe that we are assuming D 6= 0, y0 > 0, R2 > R1, respectively
corresponding to
f m 6= c(m+µ), c(m+µ−ν)> m f , c(m+µ+ν)> f m.
There are four possible positions for the straight line (3.5), which may lead however to further subcases.
4 B. Bonafe, A. Conchin Gubernati, G. Ricci, E. Venturino
Fig. 1 An example: the hyperbola with y0 > 0. For the discussion of the possible intersections with the straight line,
see the main body of the paper.
Fig. 2 A second example: another hyperbola with y0 > 0. Here for the straight line indicated by r2 in the body of the
paper, either no intersection in the first quadrant exists or exactly one is shown in the picture, but there could be two or
none if Z > R2. The straight line r4 instead is shown to have two feasible intersections if R1 < Z < R2.
1. r1: in this case there is least one feasible intersection with the hyperbola, for j > 0, w > 0, i.e.
ce f K > aλ, aλ > cA. (3.9)
2. r2: there is exactly one intersection in the first quadrant for j < 0, w > 0 and R1 < Z < R2, i.e.
ce f K < aλ,m
ec< K
cA−aλ
ce f K −aλ<
A
e f. (3.10)
But further, for Z < R1 there is no intersection in the first quadrant, while there are two, not shown in Figure 1, for Z > R2
ce f K < aλ, KcA−aλ
ce f K −aλ>
A
e f. (3.11)
3. r3: for j < 0, w < 0, i.e. ce f K < aλ, aλ < cA, there are no feasible intersections.
International Journal of Biomathematics and Systems Biology 5
Fig. 3 Another example: the hyperbola with y0 < 0. In this case we show only r1. There is no feasible intersection for
Z < R1, exactly one for R1 < Z < R2, two instead if Z > R2.
4. r4: this case is not shown in Figure 1, as it leads to several subcases, depending mainly on the slope of the straight line. In general
there could be two, one or no intersection in the first quadrant. We do not examine it further.
In order that the intersection be feasible, one must further require that F(ν)10 ≥ 0.
Similar considerations can be made on Figures 2 and 3.
Note that this equilibrium in the particular case ν = 0 can be explicitly evaluated, solving for V and F as functions of R from the
last two equations; substituting into the first one we find R, which finally leads to
E10 =
(K(λa+ f m− c(m+µ))
λa,0,
λa(m+µ)− e f K(λa+ f m− c(m+µ))
λ2a,
ecK(λa+ f m− c(m+µ))−mλa
λ2a
)
which is feasible for the following nonempty set of parameter values, as it will be seen in Section 6:
λa+ f m > c(m+µ), λa(m+µ)> e f K(λa+ f m− c(m+µ)), mλa < ecK(λa+ f m− c(m+µ)). (3.12)
The symmetric equilibrium E(ν)11 can again be obtained with a similar procedure, solving for F as a function of U in the fourth
equilibrium equation,
F =A− egU
λ, (3.13)
then substituting into the second one we obtain once again a straight line,
V =degH −bλ
HgλU +
bλ−dA
gλ.
From this and (3.13) we find the following necessary conditions for the feasibility of the equilibrium:
U <A
eg; and either U > H
dA−bλ
degH −bλ, for: degH > bλ, or U < H
dA−bλ
degH −bλ, for: degH < bλ. (3.14)
From the third equilibrium equation we then obtain the hyperbola
−e2gdU2 + egλUV +(egm+ edA)U +λ(−m−µ)V −mA = 0,
and the analysis follows the previous pattern and is therefore omitted. Once again, for ν = 0 it can be explicitly evaluated, solving for
V and F as functions of U from the last two equilibrium equations, and substituting into the second one to find U so that, finally,
E11 =
(0,
H(λb+gm−d(m+µ))
λb,
λb(m+µ)− egH(λb+gm−d(m+µ))
λ2b,
edH(λb+gm−d(m+µ))−mλb
λ2b
)
and it is feasible for the following conditions which are nonempty as shown in Section 6: