Download Impacts of maximum sustainable yield policy to prey–predator systems

Ecological Modelling 250 (2013) 134–142
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Ecological Modelling
journal homepage: www.elsevier.com/locate/ecolmodel
Impacts of maximum sustainable yield policy to prey–predator systems
T.K. Kar ∗ , Bapan Ghosh
Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah 711 103, India
a r t i c l e
i n f o
Article history:
Received 14 September 2012
Received in revised form 9 November 2012
Accepted 12 November 2012
Available online 13 December 2012
Keywords:
Prey–predator
Intraspecific competition
Ratio-dependent
Maximum sustainable total yield
Persistence
a b s t r a c t
This article investigates the effects of reaching the maximum sustainable yield (MSY) in prey–predator
systems where the prey population follows logistic law of growth. Two different models are proposed:
(i) first model involves linear prey–predator interaction and intraspecific competition among predator
populations, and (ii) the second one is a ratio-dependent prey–predator system. In the first model, our
results suggest that the introduction of intraspecific competition among predator population has important consequences for the fishing to reach MSY from prey species and maximum sustainable total yield
(MSTY) for combined harvesting of both prey and predator species. On the otherhand, in the second
model, our results suggest that though the harvesting of prey species at MSY level shall be guaranteed
the coexistence of both the species, but the combined harvesting of both the species at MSTY level may
cause extinction of the predator species. However, for both the models, predator harvesting at MSY level
may be a sustainable fishing policy. Therefore, based on our results we can conclude that MSY (or MSTY)
policy in prey–predator systems in nature are not likely to fit requirements of Conservation of Biological
Diversity (CBD, 1992) in all cases.
© 2012 Elsevier B.V. All rights reserved.
1. Introduction
The exploitation of biological resources and harvesting of population are commonly practiced in fisheries, forestry, and wildlife
management. In particular, the management of a fishery is a decision with multiple objectives. Some of the desirable objectives are
the provision of the good bio-mass yield, the conservation of fish
population, the provision of good economic returns and the provision of recreation. The formulation of good harvesting policies
which take into account these objectives is a complex and difficult
task. Maximum sustainable yield (MSY) is a simple way to manage
resources taking into consideration that over-exploited resources
lead to a loss in productivity. But, we have seen that the traditional MSY approach to fishery management relies on single species
model of population dynamics and ignores species interactions
with other member of the communities. Single species assessments
and management controls may produce misleading prediction and
pathological changes in the ecosystems. Even simple food-chain
models predict stronger compensatory responses for any species
when interactions with the rest of the system are accounted for,
owing to the negative impact of harvesting on its predators, and
the positive impacts on its food organisms.
∗ Corresponding author. Tel.: +91 3326684561; fax: +91 33 26682916.
E-mail addresses: [email protected] (T.K. Kar), keshab [email protected]
(B. Ghosh).
0304-3800/$ – see front matter © 2012 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.ecolmodel.2012.11.015
We have come through a long way from targeting MSY using
equilibrium population level. Schaefer (1954) was first to introduce MSY policy for a single species fishery having logistic law of
growth and subject to proportional harvesting. Clark (1990) also
discussed the importance of the concept of MSY policy for fishery management. Recently, Kar and Matsuda (2007) investigated
the MSY policy for single species fishery with strong Allee effect.
Legovic (2008) mentioned the effects of harvesting of MSY policy in a single species fishery. Walters et al. (2005) show that the
widespread application of single species model MSY policy would
in general cause severe deterioration in ecosystem structure, in particular the loss of top predator species. All the above results suggest
that MSY exists for a single isolated population living in nature but
there are significant dangers of adopting single-species management approaches that may be myopic. Therefore, can we really
get away with considering exploitation of one species at a time,
and ignoring interactions among species? This is the subject of our
study in this paper.
Individual species do not live in isolation, which means that
the population dynamics of different species are inevitably linked.
Therefore, it is not at all obvious whether we should expect higher
or lower yields from an entire ecosystem than would be predicted
from application of single species harvest control policy. Recently,
Matsuda and Abrams (2006) examined various food web models,
such as two-prey one predator system, three species with three
trophic levels, six species systems with multiple trophic levels and
asserted that independent harvest of species within the web can
cause extinction, but that this is less likely to happen. In their work
T.K. Kar, B. Ghosh / Ecological Modelling 250 (2013) 134–142
(2010b), Legovic and Gecek investigated a community of independent and logistically growing populations under a common
harvesting effort which leads to the maximum sustainable total
yield (MSTY). Their results suggest that in the case of two independent populations with approximately equal carrying capacities,
both the populations persist at its MSTY level. Their results also
suggest that the MSTY with a common harvesting effort implies
suboptimal fishing of some population, overfishing of others and
extinction of the rest of the populations. Legovic et al. (2010a) show
that approaching MSY in ecosystem means that most likely fish
species will be driven to extinction in every fishery that includes
exploitation of at least one trophic level which is directly or indirectly used as food for a higher trophic level. In addition, Legovic
and Gecek (2012) studied the impact of MSY policy in mutualistic
communities and interestingly noticed that harvesting of all species
to the MSTY level will induce extinction of the species with lower
biotic potentials and carrying capacities.
In this paper, we have studied the impacts of MSY policy on two
different prey–predator systems under different harvesting scenarios. Section 2 provides the general understanding about MSY policy
and revision of the outcomes due to Legovic et al. (2010a). In Section
3, we introduce an intraspecific competition term to the predator
equation and study the impacts of MSY policy when either prey or
predator species is harvested or MSTY policy when the combined
harvesting effort of both prey and predator species are taken into
account. In Section 4, we study the impacts of MSY (and MSTY) policy in a ratio dependent prey–predator system. Section 5 provides
some concluding remarks of our investigated models.
2. Effect of MSY policy in a simple prey–predator system
We first consider a single species population model followed by
proportional harvesting as follows:
x
dx
= rx 1 −
dt
k
− ex,
(1)
where x is the population size at any time t, r is the intrinsic growth
rate, k is the environmental carrying capacity of the population
and e is the harvesting effort. A basic assumption of most of the
models is to determine a catch level rate for which a stock can be
sustained indefinitely based upon the average productivity of the
stock. Therefore, largest average catch that can be employed on a
sustainable basis from a stock under existing environmental conditions has become the main goal to manage exploited populations.
The maximum sustainable yield (MSY) is based on equilibrium
biomass of the species and yield. For a single species model as given
in Eq. (1), MSY = rk/4 and it is occurred at effort level e = r/2. In fact,
even for this single species fishery, fishing at the MSY level does
not ensure constant catches in the future, because of the substantial variability in reproductive success and recruitment. In good
years, fishers may prosper with MSY based catches, but in years
when the environment is less favorable and recruitment and productivity decline, the stock will diminish and MSY may quickly lead
overfishing.
Most management measures are directed at individual stock
of a single species and these do not take into account species
interactions, such as predator-prey relationship. To study the consequences of MSY policy in multispecies system, Legovic et al.
(2010a) considered the following prey–predator system:
x
dx
= rx 1 −
dt
k
− axy
(2)
dy
= axy − my,
dt
where x and y are respectively the prey and predator biomass at
any time t. r is a constant intrinsic growth rate (biotic potential)
135
and k is the environmental carrying capacity of prey population, a
is the predation rate, m is the natural mortality rate of the predator.
The prey–predator model (2) is simple in the sense that predator
consumes prey population according to Holling type I functional
response and the intraspecific competition of the predator is not
taken into account. Introducing proportional harvesting to either
prey or predator or both the populations with combined fishing effort
we have the following results proposed by Legovic et al. (2010a) as:
In any prey–predator system, fishing to reach MSY of the prey
population only will cause extinction of the predator population.
In any prey–predator system, fishing to reach MSY of the predator population only, is unlikely to causes extinction of either prey
or predator species.
In any prey–predator system subject to equal fishing effort on
both prey and predator populations, the ultimate MSTY will be the
MSTY of a single species population composed of prey only, which
means that the predator population has gone to extinction.
3. Impacts of MSY policy under intraspecific competition
Legovic et al. (2010a) extends our knowledge on effects of MSY
policy in prey–predator models, however, they have considered
predation as the only interaction in their prey–predator systems.
But there are various other factors that may be taken into account
when modeling of such a system. One such key and somewhat novel
feature is the intraspecific competition in the predator growth
dynamics (Kuang et al., 2003; Ruan et al., 2007). This intraspecific competition is assumed to induce additional instantaneous
deaths to the predator population and the increased death rate is
proportional to the square of the predator density. To completely
eliminate density-independent mortality is as biologically unrealistic as eliminating density-dependent mortality (Caswell and
Neubert, 1998). Recently, Hixon and Jones (2005) also found this
density-dependent mortality in demersal marine fishes, which is
often caused by the interplay of predation and competition. As this
intraspecific competition enhances the mortality rate of the species,
it is expected that the MSY policy will be significantly differ from
the MSY policy in a traditional prey–predator system, and hence, in
this section, we proceed to study the MSY policy in a prey–predator
system having intraspecific competition in the predator growth
dynamics when either prey or predator or both the prey and predator populations are subject to harvest.
Therefore, our modified model becomes
dx
x
= rx 1 −
dt
k
− axy
(3)
dy
= axy − my − y2 ,
dt
where is the coefficient of intraspecific competition.
Coexistence equilibrium of the system (3) is P(x*, y*), where
x∗ = k
and
y∗ =
am + r
a2 k + r
r(ak − m)
a2 k + r
,
provided ak > m. Since the environment cannot support a population size above its carrying capacity, therefore x* < k would be
ecologically meaningful prey abundance.
We now examine whether MSY or MSTY policy can be implemented in our prey–predator system for a possible sustainable
fishing activity. Various scenarios associated with MSY and MSTY
are observed successively.
136
T.K. Kar, B. Ghosh / Ecological Modelling 250 (2013) 134–142
3.1. Prey harvesting
5
4.5
Suppose the target species is the prey only. Then after introducing the proportional harvesting of prey species the system (3)
becomes
3.5
− axy − ex
(4)
dy
= axy − my − y2 ,
dt
where e is the harvesting effort. For simplicity we consider the
catchability coefficient as unity.
The coexistence equilibrium for a fixed effort is P1 (x1∗ (e), y1∗ (e)),
where
x1∗ (e)
=k
and
y1∗ (e) =
am + r − e
a2 k + r
ak(r − e) − mr
a2 k + r
r − e
r
2
1
0
y*1
0
e
MY
0.5
1
e
.
>0
re(ak − m − e)
.
a2 k + r
If the predator is driven to extinction, then the yield is obtained as
Y (e) =
yield curve
2.5
0.5
if the effort applied to the fishery is less than the biotic potential of
prey species.
The yield function at equilibrium is
Y (e) = ex1∗ =
MY
3
1.5
Here it is observed that both prey and predator populations at
equilibrium depend linearly on employed effort. Also x1 *(e) is positive if e < r + am/ and y1 *(e) is positive if e < r − mr/ak. Hence the
coexistence equilibrium exists if the effort e < r − mr/ak. Increasing
of effort diminishes both prey and predator biomass, and ultimately
predator biomass reduces to zero if e = r − mr/ak, even though the
prey population persists.
There exists another equilibrium R(x̂1 (e), 0), where
x̂1 (e) = k
biomass, yield
1
4
m
(am + r).
a
Since Y is a quadratic function of the effort e, there is a possibility to have a single maximum. Therefore MSY may be available
from prey exploitation. We explicitly calculate the effort, eMSY for
which MSY is reached. The fishing policy that maximizes Y must be
satisfied
dY/de = 0 and d2 Y/de2 < 0 at e = eMSY ,and MSY is obtained at
eMSY = (r + am)/2.
Now if eMSY > r − mr/ak, then the predator species goes to extinction to reach the MSY level. It is difficult to explain analytically
whether MSY policy can be applied or not. Here we give some
examples to illustrate various possibilities.
Example 1. Select the ecological parameters as r = 2, k = 5, a = 0.25,
m = 0.9 and = 0.4 in appropriate units, then the predator goes to
extinction at e = 0.55 as shown in Fig. 1. The yield curve is increasing
for any effort lying in (0, 1), but if effort is greater than 0.55, then
the coexisting equilibrium disappears and the predator free equilibrium becomes stable. Therefore, e = 0.55 is a threshold value and
causes the change of stability behavior between the equilibrium
points P1 and R. The corresponding maximum of the yield curve
occurs when e = 1 and this effort is nothing but equal to r/2. The
maximum sustainable yield is equal to 2.5 and theoretically it is
rk/4. We may state that fishing of prey species to reach MSY of the
1.5
2
Fig. 1. Yield curve and equilibrium values of x1∗ and y1∗ as function of the effort e. MY
occurs at e = 1.
prey–predator system (4) is not a safe stock conservation process
and MSY does not exists in such multispecies system. For better
understanding of our analysis we introduce a term ‘maximum yield’
(MY) which means the maximum harvested biomass from ecosystem with the extinction of at least one species. By and large, the
MY of the prey–predator system (4) is equal to MSY of the single
species model (1).
Example 2. Let us now fix the parameters as r = 1, k = 5, a = 0.5,
m = 0.5 and = 0.4, then both prey and predator population coexist
and equilibrium densities of both the species decrease linearly for
e ∈ (0, 0.8). The yield curve is increasing and possess its maximum
as MY = 0.8 at e = 0.8 (see Fig. 2). If the effort is slightly increased
then the coexistence equilibrium disappears and only predator free
stable equilibrium exists. Within the effort interval (0.8, 1), the prey
population decreases with increasing rate unlike observed in the
Example 1. The yield curve also decreases with respect to effort.
Therefore MSY is not accessible form prey–predator system. Since,
in a single species model, MSY occurs at e = r/2, hence fishing of prey
to reach MSY level of the prey–predator system (4) is not equivalent
to reach MSY associated with the system (1). If it is, then MSY would
2
1.8
*
1
x
1.6
1.4
biomass, yield
dx
x
= rx 1 −
dt
k
x*
1.2
MY
1
0.8
0.6
yield curve
0.4
e
MY
0.2
y
0
0
0.2
0.4
*
1
0.6
0.8
1
e
Fig. 2. Yield curve and variation of equilibrium values of prey and predator populations. Maximum yield occurs when e = 0.8 causing the extinction of predator species.
T.K. Kar, B. Ghosh / Ecological Modelling 250 (2013) 134–142
137
4.5
x*
4
1
*
x2
3.5
yield curve
biomass, yield
biomass, yield
3
2.5
MSY
2
1.5
1
MSY
0
0.5
1
e
1.5
e
Example 3. Now we consider r = 2, k = 100, a = 0.5, m = 0.5 and
= 0.4, then the condition (r + am)/2 < r − mr/ak is satisfied.
YMSY = 2.67 occurs for eMSY = 1.33 with stable coexistence equilibrium at (2, 1.28) (see Fig. 3). The term ‘sustainable fishing’
guarantees the persistence of all species at the expanse of effort
to achieve MSY.
From the above Examples 1–3, we notice that fishery agency
should take advice before applying the MSY policy while harvesting
prey population only. It is also observed that the success of fishing
to reach MSY level depends on the mortality rate of predator fish,
biotic potential of prey, prey carrying capacity and other ecological
parameters.
3.2. Predator harvesting
− axy
(5)
dy
= axy − my − y2 − ey.
dt
The coexisting equilibrium is
=k
and
=
am + r + ae
a2 k + r
r(ak − m − e)
a2 k + r
0
Fig. 4. Yield curve and equilibrium values of x2∗ and y2∗ as function of effort. MSY can
be obtained.
This interior equilibrium exists when e ∈ (0, (ak − m)). From x2∗ (e)
and y2∗ (e) it is observed that prey population at equilibrium
increases with effort while predator population at equilibrium
decreases and goes to extinction when e = (ak − m). However, the
prey biomass does not crosses its carrying capacity as in principle
x2∗ (e) → k when predator goes to extinction. The yield at equilibrium is given by
Y=
re(ak − m − e)
.
a2 k + r
Now
dY/de = 0
and
d2 Y/de2 = − r/(a2 k + r) < 0
at
e = (ak − m)/2) < (ak − m). Hence the yield curve has a single
maximum in presence of both the species as shown in Fig. 4 and
we have eMSY = (ak − m)/2. The following theorem exhibits the
impact of fishing to reach the MSY from predator.
Theorem 1. In system (5), fishing to reach MSY from the predator
population is a sustainable fishing policy.
The effects of applying the combined harvesting effort are investigated in the following section.
3.3. Both prey and predator harvesting
If both the species are subjected to proportional harvesting with
equal effort, then the system (3) becomes
Let us consider the target species is the predator species only.
Then introducing a proportional harvesting, system (3) becomes
MSY
e
Now we shall attempt to analyze the effect of fishing on prey
population to reach MSY when (r + am)/2 < r − mr/ak holds. In this
case, yield curve attends its maximum value at eMSY = (r + am)/2
and this effort is less than the effort which causes the extinction
of the predator population. Therefore, MSY policy can be employed
to manage both the prey and predator populations. A numerical
example is provided to support the result using a different set of
parameters.
x
dx
= rx 1 −
dt
k
0
2
occur at e = 0.5. In fact, MY from both species of the system (4) is
smaller than the MSY from the prey species of the system (1).
y2∗ (e)
*
2
y
Fig. 3. Equilibrium values of prey and predator biomass exist when fishing reaches
to maximum sustainable yield.
x2∗ (e)
yield curve
e
0.5
0
MSY
y*
1
P2 (x2∗ (e), y2∗ (e)),
where
dx
x
= rx 1 −
dt
k
.
− axy − ex
(6)
dy
= axy − my − y2 − ey,
dt
The coexisting equilibrium of the system (6) is denoted by
P3 (x3∗ (e), y3∗ (e)), where
x3∗ (e) = k
and
y3∗ (e)
=
(am + r) − ( − a)e
a2 k + r
r(ak − m) − (ak + r)e
a2 k + r
.
138
T.K. Kar, B. Ghosh / Ecological Modelling 250 (2013) 134–142
3
After some manipulations, the criteria for the coexistence of both
species is obtained as
e<r
ak − m ak + r
2.5
.
< r.
This condition produces that predator is ruined from ecosystem
before the effort reaches the biotic potential of the prey population.
In this context, we address the interesting phenomenon of
Volterra’s first principle for fishing on both the species. It was found
that the proportion of predator fish increases and the proportion
of prey fish decreases during 1st world war, while the fishing is
reduced. The principle can be stated (Pireddu and Zanolin, 2008;
Legovic, 2008) as
‘If one tries to destroy uniformly and proportionally to their
numbers the individuals of the two species, then the average of the
number of individuals of the eaten species increases and the one of
the eating species decreases’. One can easily verify this principle in
Lotka–Volterra predator–prey model.
The total yield at equilibrium becomes
Y (e) = e k
(am + r) − ( − a)e
a2 k + r
+
r(ak − m) − (ak + r)e
a2 k + r
when
eMSTY =
k + r a2 k
+r
2(k + r)
y*
1
3
eMY
0.5
0
0
0.5
1
e
1.5
2
Fig. 5. Yield curve and variation of prey and predator population at equilibrium.
Predator goes to extinction before the existence of MSTY.
4. Effect of MSY policy in a ratio-dependent prey–predator
system
In the previous sections we have analyzed the impacts of fishing to reach MSY (or MSTY) level in a prey–predator system having
linear interaction. In this section, we consider a ratio-dependent
prey–predator system as follows:
dx
x
= rx 1 −
k
dt
<0
k(am + r) + r(ak − m) 1.5
.
Our goal is to maximize the total yield as well as the conservation
of both the species.
We observe that dY/de = 0 and
d2 Y
=−
de2
biomass, yield
ak + r
2
−
axy
cy + x
(7)
dy
baxy
− my,
=
cy + x
dt
.
Analytically it is very difficult to test whether eMSTY > eend
or eMSTY < eend . If eMSTY > eend , prey species survives but predator
species goes to extinction; and maximum sustainable total yield
(MSTY) cannot be obtained. Otherwise, MSTY is achieved through
the exploitation of both prey and predator populations. These two
different scenarios are supported by the following examples.
Example 4. Let us take r = 2, k = 5, a = 0.5, m = 0.5 and = 0.4.
These parameters satisfy eMSTY > eend . Now from Fig. 5, we observe
that prey biomass increases and predator biomass decreases when
effort increases from zero and ultimately predator goes to extinction when effort is 0.88. Therefore, fishing to reach MSTY eliminates
predator population with prey density as x3∗ = 2.777. If effort is
increased further, then fishing is established only from the single species population model (1). In this case, MY is obtained at
e = 1 which is the MSY obtained from single species only. Therefore
application of MSTY rule in such prey–predator system may not be
beneficial tool from ecological aspect. It is to be noted that Volterra’s
principle has been established here.
Example 5. Reset the parameters as r = 2, k = 100, a = 0.5, m = 0.5
and = 0.9. Then we have eMSTY < eend . In this case, attempt to reach
MSTY is a good management tool to conserve both prey and predator populations, through the biomass of both the prey and predator
population have a decreasing nature as effort increases (see Fig. 6).
This observation is not found in Example 4. The equilibrium density
is P3 (5.187, 0.493) when fishing reaches at MSTY and eMSTY = 1.65.
It is also to be noted that eend < r. This example contradicts the
celebrated Volterra’s principle.
where x and y are respectively the prey and predator biomass at
any time t. r is the constant intrinsic growth rate (biotic potential)
and k is the environmental carrying capacity of prey population, a is
the predation rate. b(< 1) refers as conversion rate of prey biomass
into predator biomass, c is known as half saturation constant and
m is the natural mortality rate of the predator.
10
9
MSTY
8
7
biomass, yield
ak − m yield curve
MY
Suppose
eend = r
*
x3
x*
3
6
yield curve
5
*
y3
4
3
2
eMSTY
1
0
0
0.2
0.4
0.6
0.8
1
e
1.2
1.4
1.6
1.8
Fig. 6. Equilibrium values of both the prey and predator population decrease with
respect to combined harvesting effort. Maximum sustainable total yield can be
obtained before predator goes to extinction.
T.K. Kar, B. Ghosh / Ecological Modelling 250 (2013) 134–142
k
y2
biomass, yield
biomass, yield
x1
139
y1
MSY
x2
yield curve
MSY
e
MSY
yield curve
0
eMSY
0
0
e
0
e
Fig. 7. Prey and predator biomass at equilibrium and yield curve for varying effort.
Fig. 8. Represents the yield-effort curve and the variation of equilibrium values of
both prey and predator populations.
4.1. Prey harvesting
If prey is harvested through the rule h = ex, then the system (7)
takes the following form:
x
dx
= rx 1 −
dt
k
−
axy
− ex
cy + x
(8)
dy
baxy
=
− my,
cy + x
dt
4.2. Predator harvesting
If the target species is predator then we have
The coexisting equilibrium of the system (8) is F1 (x1 (e), y1 (e)), with
k
(bc(r − e) − (ab − m))
bcr
x1 (e) =
Theorem 2. In ratio dependent prey–predator system (8), fishing to
reach MSY of the prey population only guarantees the coexistence of
both the species.
x
dx
= rx 1 −
dt
k
−
axy
cy + x
and
We have the coexisting equilibrium F2 (x2 (e), y2 (e)), where
k(ab − m)
y1 (e) =
(bc(r − e) − (ab − m)) .
bc 2 rm
x2 (e) =
It is observed that both as x1 (e) and y1 (e) decrease with effort and
ultimately tend to zero as
Hence prey and predator population exist under prey harvesting if
(ab − m)
.
bc
e<r+
The exploited yield at equilibrium is
Y (e) =
ke
(bc(r − e) − (m − ab)) .
bcr
Obviously dY/de = 0 and
e=
1
2
r+
ab − m
bc
d2 Y/de2
<
r+
ab − m
bc
.
This demonstrates that MSY exists and we have
eMSY =
1
2
r+
ab − m
bc
.
Therefore attempt to reach MSY level is always possible and
sufficient to conserve prey–predator system (see Fig. 7). This
phenomenon is not experienced for the system (4) in any circumstances. Hence we can state this finding through the following
theorem.
k(ab − m − e)
(bcr − (ab − m) + e) .
bc 2 r(m + e)
In the absence of harvesting, we have (bcr − (ab − m)) > 0 from the
existence condition of equilibrium. Therefore, F2 (x2 (e), y2 (e)) exists
if e < (ab − m).
Predator biomass ate equilibrium is a nonlinear function of effort
and it is not straight forward to comment on its variation with
respect to effort.
The yield function
Y (e) =
= − k/r < 0 at
k
(bcr − (ab − m) + e)
bcr
and
y2 (e) =
ab − m
.
bc
e→r+
(9)
baxy
dy
=
− my − ey,
cy + x
dt
ek(ab − m − e)
(bcr − (ab − m) + e)
bc 2 r(m + e)
is again highly nonlinear function of effort. Obviously Y(0) = 0 and
Y(e) has only positive zero at e = ê = (ab − m). Also Y(0 +) >0. Hence
Y(e) has a maximum at some value of e = ē ∈ (0, ê). Therefore, both
prey and predator population can coexist while fishing increases at
MSY level. Fig. 8 exhibits a possible graphical representation of the
equilibrium biomass of prey and predator and the yield curve. The
prey biomass increases linearly and predator biomass decreases
but not linearly. Hence we can state the following theorem.
Theorem 3. In ratio dependent prey–predator system (9), fishing to
reach MSY of the predator population would guarantee sustainability
of both the species.
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T.K. Kar, B. Ghosh / Ecological Modelling 250 (2013) 134–142
4.3. Prey–predator harvesting
200
180
If both prey and predator populations are harvested simultaneously with combined harvesting effort e, then we have the
following system
(10)
dy
baxy
=
− my − ey.
cy + x
dt
The coexisting equilibrium of the system (10) is F3 (x3 (e), y3 (e)),
where
x3 (e) =
k
(bcr − (ab − m) − (bc − 1)e)
bcr
bc − 1
.
Predator population at equilibrium exists if e < (ab − m).
Hence both the species coexist if
e < min
bcr − (ab − m) bc − 1
min
bcr − (ab − m) bc − 1
, (ab − m) .
It can be conclude that
, (ab − m)
= (ab − m)
(A)
if
r > (ab − m)
and
min
bcr − (ab − m) bc − 1
,
(ab − m)
=
bcr − (ab − m) bc − 1
(B)
if
r < (ab − m).
Hence we have the following theorem for the coexistence of both
the species.
Theorem 4. Both the prey and predator populations coexist if either
of (H1) or (H2) or (H3) holds, where
(H1): bc < 1 and e < (ab − m),
(H2): bc > 1, r > (ab − m) and e < (ab − m)
and
(H3): bc > 1, r < (ab − m) and e < bcr−(ab−m)
.
bc−1
The yield at equilibrium is given by
Y (e) =
ek
[ab − m + cm − (1 − c)e][bcr − (ab − m)
bc 2 r(m + e)
+ (1 − bc)e].
80
60
0
k(ab − m − e)
(bcr − (ab − m) − (bc − 1)e) .
bc 2 r(m + e)
bcr − (ab − m) x3
100
yield curve
eMY
20
Two different cases may arise as (i) bc < 1 and (ii) bc > 1.
Case I. bc < 1.
If bc < 1, then the prey biomass at equilibrium increases with
respect to the effort. Again predator population at equilibrium
exists if e < (ab − m). Therefore, as (1 − bc) is a proper fraction, both
prey and predator populations at equilibrium coexist if e < (ab − m).
Case II. bc > 1.
In this case, prey biomass at equilibrium decreases with effort
but remains positive if
e<
3
120
40
and
y3 (e) =
y
140
axy
−
− ex
cy + x
biomass, yield
x
dx
= rx 1 −
dt
k
160
0
0.5
1
e
1.5
2
Fig. 9. Increasing effort causes the extinction of the predator population. MSTY is
no longer achieved.
The impacts of fishing to reach MSTY with equal harvesting effort
on both the populations can be investigated through the following
examples.
Example 6. If we select the parameters as r = 2, k = 100, a = 0.9,
b = 0.8, m = 0.1 and c = 1, then bc < 1 holds and both the species coexist for e < 0.62. From Fig. 9 we observe that prey biomass increases
and predator biomass decreases as fishing effort increases; and
predator biomass becomes zero at e = 0.62. Hence fishing is not
reached to MSTY level in presence of positive abundance of both
the populations. If we increase the fishing effort further, then MY is
achieved when effort is half of the biotic potential of the prey, but
predator goes to extinction. Therefore attempt of fishing to reach
MSTY level gives the extinction of predator population. MY exists
at e = 1 and is equal to MSY of the system (1).
Example 7. Let r = 1 and other parameters remain same as in
Example 6. Then bc < 1 holds and both prey and predator population coexist if e < 0.62. We observe that prey biomass increases
and predator biomass decreases as fishing effort increases; and
predator is eliminated from the system if e = 0.62. The yield curve
increases for 0 < e < 0.62, but if effort is increased further, then the
character of the yield curve is changed dramatically compare to
the above example and actually it decreases as shown in Fig. 10.
MY exists at e = 0.62 and hence MY of the system (10) is not equal
to MSY of the system (1).
From Examples 6 and 7, we observe that biotic potential
has a vital role to introduce MSTY policy in a ratio-dependent
prey–predator system while dealing with equal harvesting effort
to both the species. Fishing to reach MSTY is equal to MSY of prey
harvesting if biotic potential is sufficiently large (see Fig. 9), but
neither MSTY nor MSY exists if biotic potential is significantly small
(see Fig. 10). Examples 6 and 7 are arguing the Volterra’s Principle.
Example 8. Let us consider the biological parameters as r = 1,
k = 100, a = 0.9, b = 0.8, m = 0.1 and c = 2, then all the conditions of
(H2) in Theorem 4 are satisfied. Again both MSTY for the combined
harvesting of both the species and MSY from prey harvesting do not
exist as already found in Example 7, but the fundamental difference
is that both prey and predator populations diminish (see Fig. 11) as
effort increases in 0 < e < 0.62.
Example 9. Take the parameters as r = 0.6, k = 100, a = 0.9, b = 0.8,
m = 0.1 and c = 1.5, then all conditions of (H3) of Theorem 4
are satisfied. In this case both prey and predator populations
T.K. Kar, B. Ghosh / Ecological Modelling 250 (2013) 134–142
simultaneously are driven to extinction when effort e = 0.5. Also
biomass of both the populations decreases as effort increases. The
yield curve is concave downward with respect to effort and possesses a maximum at e = 0.12 protecting both the populations from
extinction (see Fig. 12). Hence MSTY can be pursued in such situation and Volterra’s Principle no longer works for Examples 8
and 9.
140
120
100
biomass, yield
141
80
x3
y3
60
5. Conclusions
MY
40
20
yield curve
0
0
0.2
0.4
0.6
0.8
1
e
Fig. 10. Maximum yield occurs when predator is removed from ecosystem resulting
the nonexistence of MSTY.
150
y
100
biomass, yield
3
x3
MY
50
yield curve
0
0
0.2
0.4
0.6
0.8
1
e
Fig. 11. The yield curve and the variation of equilibrium values of both prey and
predator populations.
15
y3
biomass, yield
10
x
3
MSTY
5
yield curve
0
0
0.1
0.2
0.3
0.4
0.5
e
Fig. 12. Represents the yield curve and equilibrium values of prey and predator
stock. Yield curve reaches at MSTY level.
The objective of this study is to analyze the impacts of fishing
with the aims of both yield maximization and species conservation.
It is well described by Legovic et al. (2010a) that maximum yield
may drive to species extinction from a traditional prey–predator
system. In the models considered there are more cases when the
predator will not go to extinction at MSY or MSTY. We also compared our outcomes with the results due to Legovic et al. (2010a)
and other references there in. Some potential consequences of MSY
or MSTY policy has been drown for both the models.
Introducing density-dependent mortality term into traditional
prey–predator system makes the system behavior differently in
view of species coexistence while yield is maximized. Harvesting
of prey species according to catch-pre unit-effort (CPUE) hypothesis may collapse the predator species from the system to achieve
the maximum yield from target species. This result is very much
consistent with the previous study of Legovic et al. (2010a). But the
yield from the prey population in absence of the predator species
may (Example 1) or may not (Example 2) be equal to MSY as can be
obtained from the single species population model. In some occasions, harvesting of prey species at MSY level ensures the species
coexistence, which is not observed in traditional prey–predator
system (2). The equilibrium prey biomass at MSY effort level has
decreased to exactly half of its biomass before the exploitation.
Attempt to reach MSTY from both the prey and predator populations with combined harvesting effort ensures the extinction of
predator species when density-dependent mortality term is absent
in the predator (Legovic et al., 2010a). But we observe that densitydependent mortality term does matter when we activate MSTY
policy to harvest both the species. May be MSTY will not drive
predator to extinction in a prey–predator system in nature (see
Examples 4 and 5). This result differs from the results as described
by Legovic et al. (2010a).
We also consider a ratio-dependent prey–predator model to
provide some vital impacts of harvesting to maximize biomass
yield. It is observed that both prey and predator populations may
be driven to extinction together when the prey harvesting effort
crosses some threshold value (see Fig. 7). This is quite unlike to the
case of prey–predator systems like (2) and (3). Both smaller and
higher harvesting effort produces lower yield from the prey species
and in between these efforts we can set up the effort which maximize the harvested biomass. Therefore, harvesting of prey species
to reach MSY level must prevent extinction of the predator species
and it removes the uncertainty of the species coexistence for prey
harvesting as stated in our prey–predator system (3). Two folded
impacts (extinction and persistence of predator) of harvesting both
the species from the system to obtain MSTY are also discussed.
There is a possibility to conserve predator species from extinction
to reach MSTY level. It is also observed that the existence of MSTY
indicates that both the species die out with a common threshold
effort which is unlike the case in traditional prey–predator system
(see Fig. 12).
Harvesting of both the species, with increasing effort, reduces
the predator biomass and increases the prey biomass at equilibrium, which is termed as Volterra’s principle, can be found in
Lotka-Volterra model (Legovic, 2008). Our systems may agree with
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T.K. Kar, B. Ghosh / Ecological Modelling 250 (2013) 134–142
the Volterra’s principle when predator goes to extinction as yield
approaches to MSTY (see Figs. 5, 9 and 10), but it may contradict
the Volterra’s principle when MSTY exists in both the models (see
Figs. 6 and 12).
This study also gives us three potential open questions in our
future research direction which we would like to include in our
next work.
(i) Whether the above prey–predator system can be applied to
a food chain or food web model consisting of more than two
species?
(ii) Are Legovic et al.’s (2010a) results for one-prey one-predator
system are well established tool to implement into the food
chain or food web model as they already pointed out?
(iii) Is harvesting of both the prey and predator species under combined harvesting effort always sufficient to prevent extinction
of the prey species?
Acknowledgements
The research work of T. K. Kar is supported by the University
Grants Commission (UGC), India (F. No. 40-239/2011(SR), dated
29th June, 2011) and the research work of Bapan Ghosh is financed
by the Council of Scientific and Industrial Research (CSIR), India
(File No. 08/003(0077)/2011-EMR-I, dated 23rd March, 2011). The
authors are really grateful to the anonymous referees for their careful reading, constructive comments, and helpful suggestions.
References
CBD, 1992. Convention on Biological Diversity. UNEP, Nairobi. http://www.cbd.int/
(accessed 11.11.10).
Caswell, H., Neubert, M.G., 1998. Chaos and closure terms in plankton food chain
models. Journal of Plankton Research 20 (9), 1837–1845.
Clark, C.W., 1990. Mathematical Bioeconomics: The Optimal Management of Renewable Resources, 2nd ed. John Wiley and Sons, New York.
Hixon, M.A., Jones, G.P., 2005. Competition, predation, and density dependent mortality in demersal fishery. Ecology 8b, 2847–2859.
Kar, T.K., Matsuda, H., 2007. Sustainable management of a fishery with a strong Allee
effect. Trends in Applied Science Research 4, 271–283.
Kuang, Y., Fagan, W., Loladze, I., 2003. Biodiversity, habitat area, resource growth rate
and interference competition. Bulletin of Mathematical Biology 65, 497–518.
Legovic, T., 2008. Impact of demersal fishery and evidence of the Volterra principle
to the extreme in the Adriatic Sea. Ecological Modelling 212, 68–73.
Legovic, T., Klanjscek, J., Gecek, S., 2010a. Maximum sustainable yield and species
extinction in ecosystems. Ecological Modelling 221, 1569–1574.
Legovic, T., Gecek, S., 2010b. Impact of maximum sustainable yield on independent
populations. Ecological Modelling 221, 2108–2111.
Legovic, T., Gecek, S., 2012. Impact of maximum sustainable yield on mutualistic
communities. Ecological Modelling 230, 63–72.
Matsuda, H., Abrams, P.A., 2006. Maximal yields from multispecies fisheries systems: rules for systems with multiple trophic levels. Ecological Applications 16,
225–237.
Pireddu, M., Zanolin, F., 2008. Chaotic dynamics in the Volterra predator-prey model
via linked twist maps. Opuscula Mathematica 28 (4), 567–592.
Ruan, S., Ardito, A., Ricciardi, P., DeAngelis, D.L., 2007. Coexistence in competition
models with density-dependent mortality. Comptes Rendus Biologies, 845–854.
Schaefer, M.B., 1954. Some aspects of the dynamics of populations important to the
management of commercial marine fisheries. Bulletin Inter-American Tropical
Tuna Commission 1, 25–56.
Walters, C.J., Christensen, V., Martell, S.J., Kitchell, J.F., 2005. Possible ecosystem
impacts of applying MSY policies from single-species assessment. ICES Journal
of Marine Science 62, 558–568.