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BioSystems 95 (2009) 188–199
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BioSystems
journal homepage: www.elsevier.com/locate/biosystems
Effect of disease-selective predation on prey infected by contact
and external sources
Krishna pada Das a , Shovonlal Roy b , J. Chattopadhyay a,∗
a
b
Agricultural and Ecological Research Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 700108, India
School of Mathematics, University of Manchester, Manchester M13 9PL, UK
a r t i c l e
i n f o
Article history:
Received 14 May 2008
Received in revised form 15 October 2008
Accepted 16 October 2008
Keywords:
Infection by contact
External infection
Predator–prey
Disease-selective predation
a b s t r a c t
We propose and analyze a simple mathematical model for susceptible prey (S)–infected prey (I)–predator
(P) interaction, where the susceptible prey population (S) is infected directly from external sources as
well as through contact with infected class (I) and the predator completely avoids consuming the infected
prey. The model is analyzed to obtain different thresholds of the key parameters under which the system
exhibits stability around the biologically feasible equilibria. Through numerical simulations we display
the effects of external infection and the infection through contact on the system dynamics in the absence
as well as in the presence of the predator. We compare the system dynamics when infection occurs only
through contact, with that when it occurs through contact and external sources. Our analysis demonstrates
that under a disease-selective predation, stability and oscillations of the system is determined by two key
parameters: the external infection rate and the force of infection through contact. Due to the introduction
of external infection, the predator and the prey population show limit-cycle oscillations over a range
parametric values. We suggest that while predicting the dynamics of such an eco-epidemiological system,
the modes of infection and the infection rates might be carefully investigated.
© 2008 Elsevier Ireland Ltd. All rights reserved.
1. Introduction
Species are prone to infection through several sources. Recently,
a number of mathematical models have been developed and analyzed to explore predator–prey interactions with such infected
individuals. Infected prey leads to an array of interesting dynamics
of ecological interaction studied by many authors (Dobson, 1988;
Anderson and May, 1986; Chattopadhyay and Arino, 1999; Xiao and
Chen, 2001; Herbert et al., 2004). Those models have extensively
explored the dynamics of infection transmitted through contact of
species. However, many diseases are transmitted in the prey not
only through contact, but also directly from environment (e.g. several air-borne disease such as, influenza, bird flu). The sources of
external disease may be pollutants and toxicant that have a significant impact on ecological communities. Some examples are oil
pollution in the seas (Nelson, 1970) and dumping of toxic waste
in rivers and lakes (Haas, 1981; Jensen and Marshall, 1982). In a
series of papers (Luna de and Hallam, 1987; Hallam. and Clark,
1982; Hallam et al., 1983) the effects of the toxicants have been
studied on various ecosystems by analyzing mathematical models.
∗ Corresponding author. Tel.: +91 33 25753231; fax: +91 33 25773049.
E-mail addresses: shovonlal [email protected] (S. Roy), [email protected]
(J. Chattopadhyay).
0303-2647/$ – see front matter © 2008 Elsevier Ireland Ltd. All rights reserved.
doi:10.1016/j.biosystems.2008.10.003
External agent-borne diseases such as bird flu, air-borne disease
and influenza are common for prey species in a predator–prey system (Lin et al., 2003; Earn et al., 2002; Ferguson et al., 2003). The
pattern of disease transmission of this kind is to convert directly
some part of the prey population to an infected class. It is difficult
to get rid of such infections, just due to their nature of transmission. However, only in a few cases (Diego and Lourdes, 2001; Liza
et al., 2006) the effect of external infection has been explored in
predator–prey models.
On the other hand, in real-world predator–prey interactions
are there examples that a predator can recognize the infected
and noninfected (susceptible) prey. When an infection is introduced to the prey population, the predator may deliberately avoid
infected prey from its diet—a situation termed as disease-selective
predation (Roy and Chattopadhyay, 2005). The phenomenon of
avoidance of infected prey and consumption of the sound ones by
the predator pays little attention in eco-epidemiological aspects.
This selectivity of prey by predator is in apparent conflict with
much of epidemiological theory which is founded on the assumptions that likelihood of infection is equal among members of
a population and constant over space. However, the evolutionary biologists have long postulated that there should be fitness
advantages to animals that are able to recognize and avoid conspecifics infected with contact-transmitted disease (Kiesecker et
al., 1999).
K.p. Das et al. / BioSystems 95 (2009) 188–199
The experimental observation of Kiesecker et al. (1999) suggested that, by detecting chemical cues emanating from infected
individuals at a distance, the tadpoles of Bullfrog (Rana catesbenia) avoid conspecies carrying infectious yeast, Candida humicola.
Pfennig (2000) carried out experiments with spadefoot tadpoles
(Spea bombifrons, Spea multiplicata, and Scaphiopus couchii), by
feeding them either conspecific tadpoles or an equal mass of three
different species of heterospecific prey, all of which contained
naturally occurring bacteria. This study suggested that, by balancing the nutritional benefits of closely related prey with the cost
of parasite transmission, the prey of intermediate phylogenetic
similarity may provide the greatest fitness benefits to predators.
This is an example of an alternative way of using phylogenetic
similarity to assess disease risk for predators by detecting the parasites in prey (Pfennig, 2000). Thus, according to Pfennig (2000),
if a predator could recognize and avoid infected prey, it might
accrue the enhanced nutritional benefits of eating phylogenetically close prey while limiting risks of disease. There are further
studies in this context. For example, Parris et al. (2006) showed
that the infected northern leopard frog (Rana pippiens) tadpoles
were significantly farther than the uninfected ones from their
predator bluegill sunfish (Lepomis macrochirus), when the species
were exposed to visual and chemical cues together and both cues
were necessary to stimulate predator avoidance of infected animals. Further, Levri (1998) observed that the fish predators were
away from the infected individuals of snails (Potamopyrgus antipodarum).
An important feature regarding the trade-off between nutrition
and disease is the host’s immune system. If the immune system
of a predator learns to recognize and destroy parasites in its food
or if the nervous system of the predator has the sense to identify the infected prey, which may cause disease of the predator,
then the predator will always try to avoid the infected prey. However, in such a scenario for the persistence of the predator, there
should be more than one source of food. Human beings (Homo
sapiens) have this sense, and they always exhibit disease-selective
predation to prey populations like fish (Roy and Chattopadhyay,
2005).
Now, when there is infection in the prey, both by contact and
from external sources, the disease-selective predation can save the
predators from being affected from the infected prey. This process
can hardly impose any influence on the external infection. However, this predatory activity may significantly effect the fate of the
prey species, and the dynamics of the predator–prey system. Thus,
it is important to explore the dynamics of predator–prey systems,
where the prey population is infected both by external sources
and by contact, and the predator completely avoid the infected
prey.
In this article, we consider such a situation through a simple mathematical model. For the simplicity of treatment, we
assume that the infection from an external source is occurring
at a constant rate. As soon as a prey is infected, the predators
will avoid the prey. We analyze the whole model as a combination of several sub-models: (1) in the absence of predator and
(2) in the presence of predator. We analyze the predator-free
two-dimensional model by dividing in two subcases: (a) external disease-free system and (b) system under both external and
internal disease. We find suitable thresholds that has a biological significance to determine the stability of its equilibria points
and the equilibria points of the system with the predator. We further explore the system dynamics in the presence of predator by
dividing two subclasses: (a) external disease-free system and (b)
system under external and internal disease. Overall our approach
follows a local-stability analysis and suitable numerical simulations.
189
2. Formulation of the mathematical model
In formulation of the mathematical model for selective predation, we make the following assumptions:
(1) The prey population is divided into two classes in presence
of disease , viz. (a) susceptible class (S) and (b) infected class
(I). The susceptible class follows logistic growth with intrinsic growth rate r. Both prey populations have contribution to
the carrying capacity k1 . We assume a logistic growth for susceptible class because rate of population growth depends on
population density and at low densities the population growth
is maximal but population growth declines at higher density of
population. Further, many authors (Chattopadhyay and Arino,
1999; Xiao and Chen, 2001; Beltrami and Carroll, 1994) assume
that only susceptible prey population is capable of reproducing
and both prey have contribution to carrying capacity.
(2) A part of the susceptible prey population becomes infected at a
rate ˇ, following the law of mass action.
(3) The susceptible prey population is infected from external
sources at a constant rate ˇ0 .
(4) We also assume that infected prey do not grow, recover and
reproduce. The experiment on dinoflagellate Noctiluca scintillans (miliaris) in the German Bight by Uhling and Sahling (1992)
indicated that the cells become damaged and they neither feed
anymore nor reproduce.
(5) The infected prey population dies linearly at the rate d.
(6) The predator is not only dependent on the prey population for
its food but also has another source of food. So it is supposed
that the predator follows logistic growth with intrinsic growth
rate R and carrying capacity k2 (a similar situation in (Roy and
Chattopadhyay, 2005)).
(7) The predator population traces and deliberately avoids the
infected prey and thus consumes only the sound prey. As the
predator has alternative food, they do not face difficulty to
obtain the appropriate quantity of food needed for its growth
(a similar situation arisen in (Roy and Chattopadhyay, 2005)).
Under the above assumptions the mathematical model takes the
following form:
dS
dt
dI
dt
dP
dt
= rS 1 −
S+I
k1
− (ˇ0 + ˇI)S −
wPS
1 + mS
= ˇ0 S + ˇSI − dI
= RP 1 −
P
k2
+ε
⎫
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎭
wPS
1 + mS
(1)
System (1) has to be analyzed with the following initial conditions:
S(0) > 0,
I(0) > 0,
P(0) > 0,
where S, I, P are respectively the susceptible prey population,
infected prey population and the predator population. The parameter used in the model are described in Table 1.
2.1. Model in absence of predator
We first analyze the predator-free model.
dS
dt
dI
dt
S+I
= rS 1 −
k1
= ˇ0 S + ˇSI − dI
⎫
⎬
− (ˇ0 + ˇI)S ⎪
⎪
⎭
Case 1. External disease-free system (i.e. ˇ0 = 0).
(2)
190
K.p. Das et al. / BioSystems 95 (2009) 188–199
and r > (S ∗ r/k1 ) + (wP ∗ /1 + mS ∗ ), P ∗ = k2 + (k2 wS ∗ /R(1 + mS ∗ ))
The fifth and most interesting equilibrium point (from biological
point of view) is E ∗ (S ∗ , I ∗ , P ∗ )
Table 1
The symbol of parameters used in system (1).
Parameters
Significance of parameter
r
k1
ˇ
ˇ0
w
m
d
R
k2
Intrinsic growth rate of susceptible prey
Carrying capacity for both prey populations
Internal infection rate due to contact of the species
External infection rate
Grazing rate for susceptible prey
Half saturation constant
Death rate of infected prey
Intrinsic growth rate of predator
Carrying capacity for predator population
Conversion efficiency
dS
dt
dI
dt
= rS 1 −
S+I
k1
Theorem 3. (a) The equilibria points E0 (0, 0, 0), E1 (k1 , 0, 0),
E2 (S, I, 0) are unstable for all parameters values. (b) The equilibrium
point E2 (0, 0, k2 ) is locally asymptotically stable if R2 < 1 and unstable
if R2 > 1, where R2 = (r/wk2 ) (for proof see Appendix).
Theorem 4. The interior equilibrium point E ∗ (S ∗ , I ∗ , P ∗ ) is locally
asymptotically stable if the following conditions hold as follows:
⎫
⎬
− ˇIS ⎪
(3)
Theorem 1. (a) The equilibrium point E0 (0, 0) exists for all parameters values and is unstable for all parametric values. (b) If R0 > 1 then
E1 (k1 , 0) is locally asymptotically stable and if R0 < 1 then E1 (k1 , 0)
is unstable, where R0 = (d/ˇk1 ). (c) The interior point E ∗ (S ∗ , I ∗ ) is
locally asymptotically stable for all parameters values. (d) The interior
point E ∗ (S ∗ , I ∗ ) is globally stable for all parameters values (for proof
see Appendix).
Case 2. With external infection (i.e. ˇ0 =
/ 0). We consider the
model involving external and internal force of infection
S+I
= rS 1 −
k1
Proof.
The system (3) has at most three equilibrium points are E0 (0, 0),
E1 (k1 , 0), and E ∗ (S ∗ , I ∗ ), where S ∗ = (ˇ/d), I ∗ = ((ˇk1 − d)r/ˇ(r +
ˇk1 )), where ˇk1 > d. The local stability each equilibrium point of
(3) is discussed through the following theorem.
dS
dt
dI
dt
⎫
⎬
− (ˇ0 + ˇI)S ⎪
(4)
⎪
⎭
= ˇ0 S + ˇSI − dI
V=
Theorem 2. (a) If R1 < 1 then E0 (0, 0) is locally asymptotically stable and if R1 > 1 then E0 (0, 0) is unstable for the system (4), where
R1 = (r/ˇ0 ). (b) The system (4) is asymptotically stable around the
interior point equilibrium point E ∗ if R0 < (k1 − (r/ˇk12 ))S ∗ . and if
R0 > (k1 − (r/ˇk12 ))S ∗ then system (4) is unstable around the interior
equilibrium point, where R0 = (d/ˇk1 ). And if R0 = (k1 − (r/ˇk12 ))S ∗
then E ∗ may be either a center or a spiral point for system (4)(for proof
see Appendix).
2.2. In the presence of predators
Case 1. External disease-free system (i.e. ˇ0 = 0)
S+I
k1
− ˇIS −
wPS
1 + mS
= ˇSI − dI
= RP 1 −
P
k2
+ε
−
wmP ∗ S ∗
(1+mS ∗ )2
>0
wPS
1 + mS
⎫
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎭
(5)
The system of equation
has five equilibria,
E0 (0, 0, 0), E1 (0, 0, k2 ),
d r(ˇk1 − d)
E2 (k1 , 0, 0), E3
0 , where ˇk1 > d, E ∗ (S ∗ , I ∗ , P ∗ ),
ˇ ˇ(r + ˇk1 )
where S ∗ = (d/ˇ), I ∗ = (k1 /r + ˇk1 )r − ((S ∗ r/k1 ) + (wP ∗ /1 + mS ∗ ))
The Jacobian matrix at E ∗ (S ∗ , I ∗ , P ∗ ) is given by
A11
A21
A31
A12
A22
A32
A13
A23
A33
3 + ˝1 2 + ˝2 + ˝3 = 0.
The values of ˝i , i = 1, 2, 3 are given by
˝1 = −(A11 + A33 )
˝2 = (−A13 A31 + A11 A33 − A21 A12 )
˝3 = A21 A21 A33
There are at most two
equilibrium points E0 (0, 0) and
= rS 1 −
RP ∗
k2
Here
A11 = −(rS ∗ /k1 ) + (wmP ∗ S ∗ /(1 + mS ∗ )2 ), A12 =
∗
∗
−(rS /k1 ) − ˇS , A13 = −(wS ∗ /1 + mS ∗ ), A21 = ˇI, A22 = 0, A23 =
0, A31 = (wP ∗ /(1 + mS ∗ )2 ), A32 = 0, A33 = −(RP ∗ /k2 ). The characteristic equation of Jacobian matrix is given by
˝1 =
E ∗ (S ∗ , I ∗ ), where S ∗ = (A + (A2 + B)/2rˇ) and A = r(ˇ0 + ˇk1 + d),
B = 4k1 drˇ(r − ˇ0 ) and I ∗ = ((r − ˇ0 )k1 − rS ∗ /r + ˇk1 ), which exists
if r > ˇ0 and (r − ˇ0 )k1 > rS ∗
dS
dt
dI
dt
dP
dt
+
(b) ˝1 ˝2 − ˝3 > 0, where ˝i (i = 1, 2, 3) are given in the theorem.
⎪
⎭
= ˇSI − dI
rS ∗
k1
(a)
rS ∗
RP ∗
wmP ∗ S ∗
+
−
k1
k2
(1 + mS ∗ )2
w2 P ∗ S ∗
RP ∗
˝2 =
+
2
∗
k2
(1 + S m)
˝3 =
ˇRP ∗ I ∗
k2
rS∗
k1
rS ∗
wmP ∗ S ∗
−
k1
(1 + mS ∗ )2
+ ˇI ∗
rS∗
k1
+ ˇS ∗
+ ˇS ∗ .
For local stability of the interior equilibrium point E(S ∗ , I ∗ , P ∗ )
the following Routh–Hurwitz criterion must be satisfied:
(i)˝1 > 0,
(ii)˝1 ˝2 − ˝3 > 0,
(iii)˝3 > 0
The criterion (iii) is obvious and criterions (i) and (ii) are satified
if the conditions of the theorem hold. Thus the system (5) is locally
asymptotically stable if the conditions of the theorem hold.
To justify above conditions of theorem we present a numerical example. We consider a hypothetical set of parameter values as
r = 0.1, R = 0.05, k1 = 50, k2 = 20, ˇ = 0.002, w = 0.003, m = 0.2,
= 0.7, d = 0.02. For these parameters values we have S ∗ = 10,
I ∗ = 24.9572, P ∗ = 22.8. Now we investigate the local stability of
interior equilibrium point E ∗ (10, 24.9572, 22.8) and for above set
of parameter values and at E ∗ (10, 24.9572, 22.8) we have ˝1 =
0.0618, ˝2 = 0.00232338 and ˝3 = 0.000113805. Now we have
(i) ˝1 = 0.0618 > 0, (ii) ˝1 ˝2 − ˝3 = 0.0000297798 > 0 and (iii)
˝3 = 0.000113805 > 0.
So, the Routh–Hurwitz criterion is satified by the above set of
parameter values. Hence the system (5) is locally asymptotically
stable around the interior equilibrium point E ∗ (10, 24.9572, 22.8)
(see Fig. 1(b)).
K.p. Das et al. / BioSystems 95 (2009) 188–199
191
Fig. 1. (a) Extinction of infected prey population for ˇ = 0.00042. (b) Stable population distribution of three species around the interior equilibrium point for ˇ = 0.002. (c)
Oscillation starts at ˇ = 0.0025. (d) Co-existing of three populations for ˇ = 0.006. (e) Extinction of both preys at ˇ = 0.006.
Case 2. With external infection (i.e. ˇ0 =
/ 0)
dS
dt
dI
dt
dP
dt
S+I
= rS 1 −
k1
wPS
− (ˇ0 + ˇI)S −
1 + mS
= ˇ0 S + ˇSI − dI
= RP 1 −
P
k2
+ε
wPS
1 + mS
We produce some basic results on the boundedness of system
(6).
⎫
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎭
(6)
Theorem 5. All the solutions of the system (6), which initiate in R3+ ,
are uniformly bounded if ε ≤ 1, where k is a positive constant, which
should be appropriately chosen.
192
K.p. Das et al. / BioSystems 95 (2009) 188–199
Proof. Let us consider the function W = S + P + I. The time derivative of the above function along the solution of (6) is
dW
S+I
wPS
P
εwPS
= rS 1 −
−
+
− dI + RP 1 −
1 + Sm
1 + Sm
dt
k1
k2
rSI
S
wPS
P
−
+ (ε − 1)
− dI + RP 1 −
= rS 1 −
1 + Sm
k1
k1
k2
S
P
≤ rS 1 −
− dI + RP 1 −
ifε < 1.
k1
k2
2
2
Hence
(dW/dt) + kW ≤ (r/k1 )(S − k1 ) − (R/k2 )(P − k2 ) +
rk1 + Rk2 , where k = min{r, R, ˇ}
Then the above expression can be written as (dW/dt) + kW ≤ L,
where L = rk1 + Rk2 .
Now applying the theory of differential inequality we get, 0 <
W (S, I, P) < (L(1 − exp(−kt))/k) + W (S(0), I(0), P(0)) exp(−kt) and
for t → ∞, we have 0 < W < kL . Hence all solutions of (6) that initiate in R3+ are confined in the region T = ((S, I, P)R3+ : W =
for any > 0. L
k
the ratio of total death rate of the infected prey to the force of
infection through contact.
Equilibria and stability: We have at most four equilibrium points
E0 (0, 0, 0), E1 (0, 0, k2 ), E2 (S, I, 0), where I = (k1 r − (ˇ0 + Sr)/r +
ˇk1 ) and S is the positive root of the equation ˇrS 2 + [ˇˇ0 − (ˇ0 (r +
ˇk1 ) + ˇk1 r + dr)]S + d(k1 r − ˇ0 ) = 0.
Now, E2 (S, I, 0) exists if ˇˇ0 > (ˇ0 (r + ˇk1 ) + ˇk1 r + dr), k1 r <
ˇ0 and k1 r > ˇ0 + Sr The equilibrium point E ∗ (S ∗ , I ∗ , P ∗ ) is the most
important from biological point of view, where S ∗ = (dI ∗ /ˇ0 + ˇI ∗ ),
P ∗ = k2 + (k2 εwS ∗ /R(1 + S ∗ m)) and I ∗ is the positive root of the
equation
AI 4 + BI 3 + CI 2 + DI + E = 0
where
2
A = −R(ˇ + md) (r + ˇ)ˇ,
+ )
B = −wk1 k2 Rˇ3 + (mR + w)dˇ3 − 2Rˇ0 (ˇ + md)(r + ˇ)ˇ
Next, we represent two important results so that both prey populations go extinct due to selective predation in the presence of
external force of infection.
Theorem 6. If the inequality P > (1/w) (r − ˇ0 ) − ((r/k1 ) + ˇ)I
holds, where P is the predator population and I is the infected prey
population then lim S(t) = 0.
Proof.
C = −wk1 k2 [3Rˇ0 ˇ2 + 2(mR + wε)dˇ0 ˇ] − Rˇ02 (r + ˇ)ˇ
+ 2Rˇ0 (ˇ + md)[ˇk1 (r − ˇ0 ) − rd − (r + ˇ)ˇ0 ]
2
+ R(ˇ + md) ˇ0 k1 (r − ˇ0 ),
dS
(S + I)
wPS
= rS 1 −
− (ˇ0 + ˇI)S −
1 + mS
dt
k1
r
wP
= S (r − ˇ0 ) −
+ˇ I−
<0
1 + Sm
k
r 1 wP
<0
+ˇ I−
if (r − ˇ0 ) −
1 + Sm
k1
r
+ ˇ I − wP
and this implies (r − ˇ0 ) −
k1
r
wP
< (r − ˇ0 ) −
+ˇ I−
< 0,
1 + Sm
k1
2
+ R(ˇ + md) [ˇk1 (r − ˇ0 ) − rd − (r + ˇ)ˇ0 ],
i.e. (r − ˇ0 ) −
r
+ ˇ I − wP < 0
k1 r
1
(r − ˇ0 ) −
+ˇ I .
i.e.P >
w
k1
Hence the theorem.
It follows from the above theorem by putting ˇ0 = 0, that in the
external disease-free system if the inequality P > 1/w[r − (r/k1 +
ˇ)I] holds then lim S(t) = 0.
It is clear that in the external disease-free system if there is no
infected prey then the susceptible prey population will be wiped
out from the dynamical system if the predator population is just
larger than the ratio of the specific growth of the susceptible prey
population and the predation rate of predator for susceptible prey.
Theorem 7. If the inequality S < (dI/ˇI + ˇ0 ) holds then lim I(t) =
0, where S is the susceptible prey population.
Proof. (dI/dt) = ˇ0 S + ˇSI − dI < 0 iff S < (dI/ˇI) + ˇ0 .
Hence the theorem. It follows from the above theorem by putting ˇ0 = 0, that
for external disease-free system if the condition S < (d/ˇ) then
lim I(t) = 0.
The above shows that in external disease-free system for selective predation the infected prey population will be wiped out from
the dynamical system if the susceptible prey population is less than
D = −wk1 k2 [3Rˇ02 ˇ + (mR + εw)dˇ02 ] + Rˇ02 [ˇk1 (r − ˇ0 )
− rd − (r + ˇ)ˇ0 ] + 2Rˇ02 (ˇ + md)k1 (r − ˇ0 ),
E = Rˇ03 k1 (r − ˇ0 ) − wk1 k2 Rˇ03 .
Here we investigate the positive roots of equation for I. For
this we use the set of parameters which are described in Case 1.
For that set of parameters values we get A = −3.672 × 10−10 , B =
−1.37806 × 10−9 , C = 7.362 × 10−8 , D = 5.139 × 10−7 , E = 6.75 ×
10−7 and the equation for I takes the form as
−3.672 × 10−10 I 4 − 1.37806 × 10−9 I 3 + 7.362 × 10−8 I 2
+ 5.139 × 10−7 I + 6.75 × 10−7 = 0.
So, by Descartes’ rule of signs there exists only one positive root
of above equation and roots of the equation are given by −11.8111,
−5.67178, −1.77032 and 15.5003.
Now corresponding the positive root I ∗ = 15.5003 we get S ∗ =
5.08202, P ∗ = 22.1171 and we get uniqe interior equilibrium point
E ∗ (5.08202, 15.5003, 22.1171).
Theorem 8. (a) The equilibrium points E0 (0, 0, 0), E2 (S, I, 0) are
unstable for all parameters values for the system (6). (b) If R1 < 1
then the equilibrium point E1 (0, 0, k2 ) is stable but when R1 > 1
then the stability of E1 (0, 0, k2 ) depends upon R2 . In that situation
E1 (0, 0, k2 ) is locally asymptotically stable if R2 > (r/r − ˇ0 ) and
R2 < (r/r − ˇ0 ) implies unstable of equilibrium point E1 (0, 0, k2 ) (for
proof see Appendix).
Theorem 9. The system (6) is locally asymptotically stable around
the interior point E ∗ (S ∗ , I ∗ , P ∗ ) the following conditions hold,
(a)
rS ∗
k1
+
P∗ R
k2
+d−
mwP ∗ S ∗
(1+S ∗ m)2
− ˇS ∗ > 0
(b) 1 2 − 3 > 0
(c) 3 > 0, where i (i = 1, 2, 3) is given in the theorem.
K.p. Das et al. / BioSystems 95 (2009) 188–199
Proof.
V=
The Jacobian matrix at E ∗ (S ∗ , I ∗ , P ∗ ) is given by
A11
A21
A31
A12
A22
A32
A13
A23
A33
2 =
The characteristic equation of Jacobian matrix is given by,
3 =
RP ∗
k2
×
The quantities i , i = 1, 2, 3 are given by
1 = −(A11 + A22 + A33 )
2 = A11 A33 + A22 A33 + A11 A22 − A12 A21 − A13 A31
3 = A13 A31 A22 + A12 A21 A33 − A33 A11 A22
1 =
rS ∗
mwP ∗ S ∗
−
k1
(1 + S ∗ m)2
+ (ˇS ∗ − d)
Here
A11 = −(rS ∗ /k1 ) + (mwP ∗ S ∗ /(1 + S ∗ m)2 ), A12 =
−(rS ∗ /k1 ) − ˇS ∗ , A13 = −(wS ∗ /1 + mS), A21 = ˇ0 + ˇI, A22 = ˇS ∗ −
d, A23 = 0, A31 = (wP ∗ /(1 + mS ∗ )2 ), A32 = 0, A33 = −(RP ∗ /k2 ).
3 + 1 2 + 2 + 3 = 0.
RP ∗
k2
193
rS ∗
mwP ∗ S ∗
RP ∗
∗
−
−
ˇS
+
d
+
k1
k2
(1 + S ∗ m)2
rS∗
k1
P ∗ S ∗ mw
(1 + mS ∗ )2
rS ∗
+
−
k1
+ ˇS ∗ (ˇ0 + ˇI)
rS ∗
P∗R
−
k1
k2
+
w 2 P ∗ S ∗
(1 + S ∗ m)3
+ ˇS ∗ (ˇ0 + ˇI) + P ∗ S ∗ (ˇS ∗ − d)
P ∗ Rmw
k2 (1 + mS ∗ )2
−
rR
εw2
+
k1 k2
(1 + mS ∗ )3
For local stability of E ∗ (S ∗ , I ∗ , P ∗ ) the Routh–Hurwitz criterion,
viz. (i) 1 > 0, (ii) 1 2 − 3 > 0 and (iii) 3 > 0 are satisfied if the
conditions of the heorem hold. Hence, under the conditions of theorem the system (6) is locally asymptotically stable around interior
equilibrium point E ∗ (S ∗ , I ∗ , P ∗ ).
To investigate the local stability of the system around the interior equilibrium point E ∗ (S ∗ , I ∗ , P ∗ ), we give a numerical example.
For this we use a set of parameter described in Case 1 of this
section and that set of parameter values we get unique interior
equilibrium point E ∗ (5.08202, 15.5003, 22.1171). For that set of
Fig. 2. (a) Prevention of infected prey population for ˇ = 0.00042 and r = 0.2. (b) Stable population distribution for r = 0.3 and ˇ = 0.0025. (c) Co-existing of three populations
in limit-cycle oscillation for ˇ = 0.015 and r = 0.3. (d) Stable distribution of three populations for r = 0.5 and ˇ = 0.015.
194
K.p. Das et al. / BioSystems 95 (2009) 188–199
parameter values and at E ∗ (5.08202, 15.5003, 22.1171) we get
1 = 0.058706, 2 = 0.00145195 and 3 = 0.000480684.
For local stability of E ∗ (5.08202, 15.5003, 22.1171) the
Routh–Hurwitz criterion viz (i) 1 = 0.058706 > 0, (ii)
1 2 − 3 = 0.0000193175 > 0 and (iii) 3 = 0.000480684 > 0
are satisfied. Thus the system is locally asymptotically statble
around E ∗ (5.08202, 15.5003, 22.1171) (see Fig. 3(b)).
Theorem 10. When the rate of internal infection (ˇ) crosses a critical
value, the system enters into Hopf bifurcation around the positive equilibrium. The necessary and sufficient conditions for Hopf bifurcation
to occur is that there exists ˇ = ˇ∗ such that,
(a) H(ˇ∗ ) ≡ 1 (ˇ∗ )2 (ˇ∗ ) − 3 (ˇ∗ ) = 0
(b)
dRe((ˇ))
dˇ
ˇ=ˇ∗
=
/ 0.
The condition 1 2 − 3 = 0 is given by H ≡ [(rS/k1 ) + (PR/k2 ) +
d − (mwPS/(1 + Sm)2 ) − ˇS] [(ˇS − d){(PSmw/(1 + mS)2 ) −
(Sr/k1 ) − (PR/k2 )} + (εw2 SP/(1 + Sm)3 )] − PS(ˇS −
d)[(RPmw/k2 (1 + mS)2 ) − {(Rr/k1 k2 ) + (εw2 /(1 + mS)3 )}] = 0
(for proof see Appendix).
3. Biological significance of the threshold parameters
We first discuss the biological significance of three threshold
parameters obtained, each of which has clear and distinct biological
meaning.
(a) R0 = (d/ˇk1 ) determines the local stability of E = (k1 , 0), the
axial equilibrium on S-axis in the external disease-free and
predator-free system (3). From this local stability analysis it is
observed that R0 determines that the system (3) will be disease free if R0 > 1 (i.e. d > ˇk1 that indicates biologically that
mortality of infected prey is higher than the prey infected from
susceptible prey). It also determines the existence of E ∗ (S ∗ , I ∗ ),
interior equilibrium point of system (3) if R0 < 1 that signifies
that mortality of infected prey is lower than the prey infected
from susceptible prey and stability of E ∗ (S ∗ , I ∗ ) interior equilibrium point of the system (4).
(b) R1 = (r/ˇ0 ) determines the local stability of E0 (0, 0), the trivial
equilibrium point of the system (4). From this local stability we
draw a result that all population of system (4) wipe out if R1 < 1
(i.e. r < ˇ0 that signifies that growth rate of susceptible prey is
lower than prey infected externally from susceptible prey). So,
the external force of infection has an important role to wipe out
all populations from the system (4). In this ecological point of
Fig. 3. (a) Stable population distribution for ˇ = 0.002. (b) Stable population distribution for ˇ0 = 0.03 and ˇ = 0.002. (c) Extinction of susceptible and infected prey
populations for ˇ0 = 0.04.
K.p. Das et al. / BioSystems 95 (2009) 188–199
view R1 can be considered as a controlling threshold to maintain the prevention of all population in the system (4). It also
determines local stability of E(0, 0, k2 ), the equilibrium point of
the system (6), where the internal and external infection exist.
(c) R2 = (r/wk2 ) determines the local stability of E(0, 0, k2 ), the
equilibrium point of the external disease-free system (5). Here r
is the growth rate of the susceptible prey w k2 is the total grazing susceptible prey at E(0, 0, k2 ). So R2 can be considered as
the ration of growth rate of the susceptible prey to the maximum grazing on susceptible prey population. When R1 > 1, R2
determines the local stability of E(0, 0, k2 ), equilibrium point of
the system (6).
4. Numerical results and discussions
195
(threshold) of internal infection (ˇmin = 0.00042) below which the
infected prey population does not persist and hence the disease
does not spread in the prey population. For (ˇ > 0.00042) there is
a range of (0.00042 < ˇ < 0.0025), where both the susceptible and
infected prey population coexist at equilibrium with their predator
population. At ˇ = 0.0025 and above which all three population
co-exist in limit-cycle oscillation and this limit-cycle oscillation
continues up to ˇ = 3.0. There is a maximum threshold of ˇ = 3
above which both preys go to extinction and predator survives.
The figures also show that the response of the prey population and
therefore the predator population to increasing force of infection
is highly nonlinear. These threshold phenomena for the force of
infection may be used as a control parameter for monitoring the
dynamics of the system.
In this section, we use the numerical results to confirm and visualize our analytical findings. The dynamics of system around the
positive interior steady state has been numerically simulated for a
wide range of parameter values. We consider a hypothetical set of
parameter values as r = 0.1, R = 0.05, k1 = 50, k2 = 20, ˇ = 0.002,
w = 0.003, m = 0.2, = 0.7, d = 0.02 (with appropriate parameters), which satisfy the existence condition and stability condition.
The specific growth rate r of the susceptible prey S and the internal
force of infection ˇ are the two parameters that directly influence
the population density of the preys. In numerical analysis we have
used the initial values (20, 10, 17) of susceptible prey, infected prey
and predator populations, respectively.
4.2. Population stability in the variation of ˇ and r parameters
4.1. Dynamics of the system (5) for increasing the rate of internal
infection ˇ
Next we introduce the external force of infection in the system where only internal force of infection is present. Keeping all
parameters values (which are used in the external disease-free system) fixed and internal force of infection ˇ = 0.002 (i.e. keeping
the internal force of infection ˇ at stable co-existing equilibrium)
The dynamics of the three populations (for r = 0.1) around E ∗
for changing ˇ is shown in Fig. 1. There is a minimum strength
Fig. 2 shows the system obtained through linear stability analysis for variation in both the force of internal infection (ˇ) and the
specific growth rate r of the susceptible prey. We observe that for
increasing ˇ and the specific growth rate r the dynamics of the
system changes. Increase of the specific growth rate r prevents
the extinction of infected prey population, and the oscillating coexistence of three populations changes to a stable co-existence
(Fig. 2).
4.3. Effect of rate of external infection (ˇ0 ) in the system (6)
Fig. 4. External disease-free system bifurcates at ˇ = 0.0025.
196
K.p. Das et al. / BioSystems 95 (2009) 188–199
Fig. 5. The system bifurcates at ˇ = 0.0035 and ˇ = 0.0085 when ˇ0 = 0.016.
Fig. 6. The system bifurcates at ˇ0 = 0.0175 when ˇ = 0.004.
K.p. Das et al. / BioSystems 95 (2009) 188–199
we observe the changeability of dynamics of the external diseasefree system in introduction of external force of infection. We obtain
an interval of ˇ0 (0, 0.04) where three populations co-exist. There
is a maximum value ˇ0 (max ˇ0 = 0.04) above which susceptible
and infected cannot survive (Fig. 3). We observe that in the stable
co-existing equilibrium position at ˇ = 0.002 the susceptible and
predator survive in the presence of infected prey in the wide range
of external infection (0, 0.04) (Fig. 3).
197
It is worth mentioning that, although we have presented a
number of real-world examples of disease-selective predation, our
study is restricted only in the extensive theoretical analysis of
such scenarios. Our theoretical study could only capture the major
mechanisms associated with an eco-epidemiological aspect where
a predator selectively avoids infected prey. Further experimental
or field investigations in this directions may be helpful to verify
whether or not the processes prescribed by the the model actually
function in a similar manner in the real world.
4.4. Bifurcation diagram
To demonstrate the above results as a function of different
parameters of interest, we have provided bifurcation diagrams.
For the external disease-free system we observe that the system
bifurcates at ˇ = 0.0025, for ˇ higher than this value the system
oscillates and for lower than this value the system stabilizes (Fig. 4).
When the external force of infection is introduced at a fixed
rate, it is the magnitude of internal infection that determines the
dynamic behaviour. For ˇ0 = 0.016, a moderate range of value of
ˇ(0.0035, 0.0085) leads to oscillation (with Hopf bifurcation at the
end points of the interval), however, if the value of ˇ is either less
than or higher than this range, the system is stabilized (Fig. 5).
On the other hand, when the value of internal infection is kept
fixed, there is a bifurcation value ˇ0 = 0.0175 above which the system is stabilized and below which it oscillates around the interior
equilibrium (Fig. 6).
Acknowledgements
This research was supported by a project fund of Indian Statistical Institute. The authors are grateful to the learned reviewer for
his critical comments and suggestions on the earlier version of the
paper.
Appendix A
Proof of Theorem 1. The Jacobian matrix of the system (3) at (S, I)
is given by
J=
2S + I
) − ˇI
k1
−
rS
− ˇS
k1
ˇS − d
ˇI
The Jacobian matrix at E0 (0, 0) is given by
5. Conclusion
Under disease-selective predation in the presence of an alternative food, the survival of a predator is less likely dependent
on the dynamics of a susceptible-infected prey community. However, this mode of predation affects significantly the dynamics of
prey population infected through contact as well as from external sources. It is noteworthy that due to inclusion of the external
disease, the present model is different from the previously used
eco-epidemiological models (e.g. Chattopadhyay and Arino, 1999;
Herbert et al., 2004). The stability analysis and the numerical
simulations in our study demonstrates a number of interesting phenomena. Firstly, if the predator is absent in the system, the disease
dynamics under the influence of external infection is significantly
different from that when the prey get infected only through contact.
In an non-external-infection scenario, the susceptible and infected
population exhibit a locally as well as globally stable dynamics.
However, under the influence of both external and contact infection, the stability and instability of susceptible and infected prey
depends on some of the key parameters, such as rate of infection through contact, mortality of infected prey and the carrying
capacity. On the other hand, when the influence of disease-selective
predation is incorporated, the species dynamics are strongly dependent on the external infection rate and force of infection through
contact. When external disease is absent, there is a maximum
threshold of internal force of infection above which both prey populations go to extinction. But, due to logistic growth of predator
does not go to extinction. However, for a fixed value of external
infection all three species co-exist through the limit cycles oscillation in an interval of force of infection through contact, and exhibit
stable dynamics both below and above that range. When the force
of infection through contact is fixed, all three populations co-exist
in the stable equilibrium position on introduction of external infection. Thus, external infection and disease-selective predation have
a influential role in the predator–prey systems with prey disease.
We suggest that cautionary attention on the both infection parameters (external and contact) might be taken while predicting the
dynamics of such eco-epidemiological systems.
r(1 −
r
k1
J=
0
0
−d
From Jacobian matrix it is clear that the equilibrium point
E0 (0, 0) is unstable.
The Jacobian matrix at E1 (k1 , 0) is given by
J=
−r
−r − ˇk1
0
ˇk1 − d
From Jacobian matrix it is obvious that the stability of E1 (k1 , 0)
depends on ˇk1 − d, i.e. (d/ˇk1 ) > 1, i.e. R0 = (d/ˇk1 ) > 1 and
unstable if R0 = (d/ˇk1 ) < 1
The Jacobian matrix is given by
⎡
⎢
J=⎣
rd
ˇk1
(ˇk1 − d)r
r + ˇk1
−
−
(r + ˇk1 )d
ˇk1
⎤
⎥
⎦
0
The characteristic equation is given by
2 + c1 +
rd(ˇk1 − d)
=0
ˇk1
where
c1 =
rd
ˇk1
and
c2 =
rd(ˇk1 − d)
ˇk1
As (ˇk1 − d) > 0, c1 , c2 are positive.
By the Routh–Hurwitz criterion, the interior equilibrium point
E(S ∗ , I ∗ ) is locally asymptotically stable for all parameters values.
f1 (S, I) = rS 1 −
f2 (S, I) = ˇSI − dI
S+I
k1
− ˇIS
198
K.p. Das et al. / BioSystems 95 (2009) 188–199
The Jacobian matrix at E1 (k1 , 0, 0) is given by
We take h(S, I) = (1/SI) then f1 h = r((1/I) − (S + I/Ik1 )) − ˇ and
(∂f1 h/∂I) = −(r/Ik1 )
f2 h = ˇ −
⎡
⎢
⎢
∂f
d
and h = 0
S
∂h
−r
0
J=⎢
⎢
Now
∂f1 h
∂f2 h
r
+
=−
<0
Ik1
∂S
∂I
which is negative in the interior of the 1st quadrant.
By Dulac criterion (negative) the system has no periodic
orbit in the 1st quadrant and E ∗ (S ∗ , I ∗ ) is globally asymptotically
stable. Proof of Theorem 2. The Jacobian matrix of the system (4) at (S, I)
is given by
J=
2S + I
) − (ˇ0 + ˇI)
k1
ˇ0 + ˇI
r(1 −
rS
− ˇS
k1
ˇS − d
−
J=
r − ˇ0
ˇ0
0
−d
J=
−r
S
k1
−
ˇ0 + ˇI
0
0
R+
⎡
r
S
k1
⎢
⎢
ˇI
J=⎢
⎢
⎣
0
⎡
−
rS
− ˇS
k1
0
R+
r − wk2
wk2
rS
− ˇS
k1
0
0
⎤
−d
0
⎦
0
−R
Proof of Theorem 8.
is given by
⎡
J=
The Jacobian matrix of the system (5) at
2S + I
wP
) − (ˇ0 + ˇI) −
2
k1
(1 + mS)
⎢ ˇI + ˇ
0
⎣
wP
2
−
rS
− ˇS
k1
−
Sw
1 + mS
ˇS − d
0
0
2P
wS
R(1 −
)+
1 + Sm
k2
⎤
⎥
⎦
The Jacobian matrix at E0 (0, 0, 0) is given by
⎡
r − ˇ0
J = ⎣ ˇ0
2S + I
wP
) − ˇI −
2
k1
(1 + mS)
wP
2
−
rS
− ˇS
k1
−
Sw
1 + mS
ˇS − d
0
0
2P
wS
R(1 −
)+
1 + Sm
k2
r
0
0
0
0
−d
0
0
R
⎤
0
−d
0
0⎦
0
R
From Jacobian matrix it is clear that E0 (0, 0, 0) is unstable.
The Jacobian matrix at E2 (S, I, 0) given by
⎡ S
rS
Sw
r
−
− ˇS −
1 + mS
k1
⎢ k1
⎢
J = ⎢ ˇI + ˇ0 ˇS − d
0
⎣
0
R+
0
⎤
⎥
⎥
⎥
⎦
The Jacobian matrix at E0 (0, 0, 0) is given by
J=⎣
Jacobian matrix of the system (6) at (S, I, P)
ˇS − d
Proof of Theorem 3.
(S, I, P) is given by
⎡
⎥
⎥
⎥
⎥
⎦
r(1 −
(1 + Sm)
By the Routh–Hurwitz criterion, the interior equilibrium point
E(S ∗ , I ∗ ) is locally asymptotically stable for all parameters values if R0 = (d/ˇk1 ) < ((1/k1 ) − (r/ˇk1 ))S ∗ and unstable if R0 =
(d/ˇk1 ) > ((1/k1 ) − (r/ˇk1 ))S ∗ . (1 + Sm)
wS
1 + Sm
⎤
From Jacobian it is clear that the equilibrium point E2 (0, 0, k2 )
is stable if r − wk2 < 0, i.e. R2 = (r/wk2 ) < 1 and is unstable if r −
wk2 > 0, i.e. R2 = (r/wk2 ) > 1. 0
⎢
⎢
J = ⎢ ˇI
⎣
Sw
1 + mS
0
d
1
r
c1 < 0 ⇒ R0 =
<(
−
)S ∗
k1
ˇk1
ˇk1
r(1 −
−
0
J = ⎣0
The characteristic equation is given by 2 + c1 + c2 = 0, where
c1 = [(rS/k1 ) − ˇS + d]c2 = ((r/k1 ) + ˇ)S(ˇ0 + ˇI)
Here, c2 > 0 c1 > 0 ⇒ R0 = (d/ˇk1 ) > ((1/k1 ) − (r/ˇk1 ))S ∗
⎡
wk1
1 + k1 m
⎥
⎥
⎥
⎥
⎦
From the Jacobian matrix it is follows that E3 (S, I, 0) is unstable.
The Jacobian matrix at E2 (0, 0, k2 ) is given by
From Jacobian matrix it is clear that the equilibrium point
E0 (0, 0) is locally asymptotically stable if r − ˇ0 < 0, i.e. (r/ˇ0 ) < 1,
i.e. R1 = (r/ˇ0 ) < 1 and unstable r − ˇ0 > 0, i.e. (r/ˇ0 ) > 1, i.e.
R1 = (r/ˇ0 ) > 1.
The Jacobian matrix at E ∗ (S ∗ , I ∗ ) is given by
ˇk1 − d
⎤
From the Jacobian matrix it is clear that E1 (k1 , 0, 0) is unstable.
The Jacobian matrix at E3 (S, I, 0) is given by
The Jacobian matrix of (4) at E0 (0, 0) is given by
−
⎣
0
k1 w
1 + mk1
−r − ˇk1
⎤
⎦
From Jacobian matrix it follows that E0 (0, 0, 0) is unstable.
wS
1 + Sm
⎤
⎥
⎥
⎥
⎦
From Jacobian matrix it is obvious that E2 (S, I, 0) is unstable.
The Jacobian matrix at E1 (0, 0, k2 ) is given by
⎡
r − ˇ0 − wk2
J = ⎣ ˇ0
wk2
0
−d
0
0
0
−R
⎤
⎦
It is clear that the stability of the equilibrium point depends
upon the eigen value r − ˇ0 − wk2 .
If r − ˇ0 < 0, i.e. R1 = (r/ˇ0 ) < 1 then E1 (0, 0, k2 ) is locally
asymptotically stable. when R1 > 1 then the equilibrium point
E1 (0, 0, k2 ) is locally asymptotically stable if r − ˇ0 < wk2 , i.e. R2 >
r/(r − ˇ0 ) and unstable if R2 < r/(r − ˇ0 ) K.p. Das et al. / BioSystems 95 (2009) 188–199
Proof of Theorem 10. The necessary and sufficientconditions for
Hopf bifurcation to occur is that there exists ˇ = ˇ∗ such that
dRe((ˇ))
dˇ
ˇ=ˇ∗
=
/ 0
References
The condition 1 2 − 3 = 0 is given by
H ≡
PR
mwPS
rS
+
+d−
− ˇS
k1
k2
(1 + Sm)2
Sr
PR
εw2 SP
× (ˇS − d){
−
−
}+
2
k
k
1
2
(1 + mS)
(1 + Sm)3
PSmw
− PS(ˇS − d)
RPmw
k2 (1 + mS)2
−{
Rr
εw2
+
} =0
k1 k2
(1 + mS)3
For ˇ = ˇ∗ we have
(2 + d2 )( + d1 ) = 0
which has three roots 1 = i d2 , 2 = −i d2 , 3 = −d1
For all ˇ, the roots are in general of the form
1 (ˇ) = d1 (ˇ) + id2 (ˇ),
2 (ˇ) = d1 (ˇ) − id2 (ˇ),
3 (ˇ) = −d1 (ˇ)
Now, we shall verify the transversality condition
dRe(j (ˇ))
dˇ
ˇ=ˇ∗
=
/ 0,
conditions hold. This implies that a Hopf bifurcation occurs at
ˇ = ˇ∗ . Hence the theorem.
(a) H(ˇ∗ ) ≡ 1 (ˇ∗ )2 (ˇ∗ ) − 3 (ˇ∗ ) = 0
(b)
199
j = 1, 2.
Substituting j (ˇ) = d1 (ˇ) + id2 (ˇ), into (1) and calculating the
derivative, we have
K(ˇ)d1 (ˇ) − L(ˇ)d2 (ˇ) + M(ˇ) = 0
K(ˇ)d1 (ˇ) + L(ˇ)d2 (ˇ) + N(ˇ) = 0
where K(ˇ) = 3d12 (ˇ) + 2d1 (ˇ)d1 (ˇ) + d2 (ˇ) − 3d22 (ˇ)
L(ˇ) = 6d1 (ˇ)d2 (ˇ) + 2d1 (ˇ)d2 (ˇ)
M(ˇ) = d12 (ˇ)d1 (ˇ) + d2 (ˇ)d1 (ˇ) + d3 (ˇ) − d1 (ˇ)d22 (ˇ)N(ˇ) =
2d1 (ˇ)d2 (ˇ)d1 (ˇ) + d2 (ˇ)d2 (ˇ).
Since
L(ˇ∗ )N(ˇ∗ ) +
/ 0, we have (dRe(j (ˇ))/dˇ)ˇ=ˇ∗ = (LN + KM/K 2 +
K(ˇ∗ )M(ˇ∗ ) =
L2 ) =
/ 0, and 3 (ˇ∗ ) = −d1 (ˇ∗ ) =
/ 0. Therefore the transversality
Anderson, R.M., May, R.M., 1986. The invasion and spread of infectious diseases
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