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Biological control through provision of additional food to predators: A theoretical study
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Theoretical Population Biology 72 (2007) 111–120 www.elsevier.com/locate/tpb
Biological control through provision of additional food to predators: A theoretical study P.D.N. Srinivasua,, B.S.R.V. Prasada, M. Venkatesulub,1 a
Department of Mathematics, Andhra University, Visakhapatnam 530 003, India Department of Mathematics & Statistics, College of Arts & Science, University of Missouri, Kansas City, Missouri-64110, USA
b
Received 29 September 2006 Available online 10 April 2007
Abstract In this paper we analyze a variation of a standard predator–prey model with type II functional response which represents predator–prey dynamics in the presence of some additional food to the predator. The aim is to study the consequences of providing additional food on the system dynamics. We conclude that handling times for the available foods play a key role in determining the eventual state of the ecosystem. It is interesting to observe that by varying the quality and quantity of additional food we can not only control and limit the prey, but also limit and eradicate the predators. In the context of biological pest control, the results caution the manager on the choice of quality and quantity of the additional food used for this purpose. An arbitrary choice may have opposite effects leading to increase in pest concentration and eradication of the predator. This study offers insight into the possible management strategies that involve manipulation of quality and supply level of additional food to predators, for the benefit of biological control. The theoretical conclusions agree with results of some practical biological control experiments. r 2007 Elsevier Inc. All rights reserved. Keywords: Biological control; Additional food; Predator; Prey; Pest; Handling time
1. Introduction The consequences of providing a predator with additional food and the corresponding effects on the predator–prey dynamics and its utility in biological control have been the topic of study for many scientists. A major portion of the literature, dealing with biological control aspects, assumes the role of pest for the prey. These studies address the controllability and eradication of the pest through the predator by providing the latter with alternative or additional food. This is modeled mathematically as one predator–two (non-interacting) prey system. Such shared predator models predict that adding additional prey (different from the one existing in the ecosystem Corresponding author. Fax: +91 891 2755324.
E-mail addresses: [email protected] (P.D.N. Srinivasu), [email protected] (B.S.R.V. Prasad), [email protected] (M. Venkatesulu). 1 Permanent address: Professor of Mathematics, Department of Computer Applications, Arulmigu Kalasalingam College of Engineering, Krishnankoil 626190, Tamil Nadu, India. 0040-5809/$ - see front matter r 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.tpb.2007.03.011
which could also be a non-reproducing prey or food source) increases predators and also their effect on target prey, thereby helping to decrease the abundance of prey (Holt and Lawton, 1994; van Baalen et al., 2001; van Rijn et al., 2002). This phenomenon, i.e., the presence of alternative food for a predator leading to reduction in equilibrium prey densities, is termed as apparent competition (Holt, 1977). A lot of theoretical work has been done in assessing the effects that can occur among the prey that share a predator. Some of these works can be found in the papers reviewed in Holt (1984) and also in the recent works of Harmon (2003). Most of these theoretical models conclude that adding a non-pest alternative prey to a predator–prey system would lower the density of the target prey. By and large, these works infer that manipulating non-target prey can influence the predator in a way that increases target predation and eventually controls the prey population. However, from empirical data it is clear that addition of alternative prey does not always increase target predation (Harwood and Obrycki, 2005; Holt and Lawton, 1994;
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Murdoch et al., 1985; Wootton, 1994). Thus, there is an apparent conflict between theory and empirical observations. In the recent works of Sabelis and van Rijn (2005), it was pointed out that alternative food need not always promote biological pest control and it depends on the role of this food, whether it is substitutable or complementary to the prey. As per the findings in Harwood and Obrycki (2005), the effect of consuming non-pest species on rates of pest predation by a generalist predator can be twofold; feeding upon these nutritious food items generally enhances fecundity thus improving their population growth, but the presence of alternative prey, especially during times when pest regulation is required, can result in reduced levels of pest consumption per individual predator. From the above observations we recognize that provision of additional food to a predator may or may not reduce the target prey density. This calls for a better understanding of the mechanisms, assumptions and biological hypothesis of predator–prey dynamic models involving additional food. Investigation in this direction is likely to improve our understanding on the consequences of providing additional food to a predator, thereby bringing out an explicit link between practical biological control and theoretical studies. In this paper, we consider a modified version of a predator–prey model with Holling type II predation to reflect the dynamics of predator and prey in the presence of some additional food for the predators. It is assumed that the additional food is not dynamic but maintained at a specific constant level either by the nature or by an external agency. The advantage of this approach is that it reduces the dimension of the system from three to two, which allows the phase plane analysis for studying the consequences of availability of additional food to the predator (van Baalen et al., 2001). This also describes ecosystem dynamics when a non-reproducing prey or food source is added to it (Holt and Lawton, 1994; van Baalen et al., 2001; van Rijn et al., 2002). In the model under consideration, it is assumed that the number of encounters per predator with the additional food is proportional to the density of additional food in the environment and the predators are not assumed to be optimal foragers. Here, we analyze the dynamics of this model in a systematic manner, study the dependance of the dynamics on some identified vital parameters and discuss the global behavior and controllability of the ecosystem. Our interest is to investigate the consequences of providing additional food to the predators on the predator–prey dynamics. From this study, it is possible to develop management strategies that manipulate quality of the additional food and its supply level for the benefit of biological control. This study illustrates that exploring the fundamental nature and ecological mechanisms of predator–food interactions, we will be in a position to understand the indirect interactions taking place in the ecosystem and their vital role in influencing the predators, which helps to predict and control the eventual state of the ecosystem and
ultimately determine prey–predator regulation (Coll and Guershon, 2002; Holt and Lawton, 1994; Wootton, 1994). In this study it is found that, by providing the predator with appropriate additional food (characterized by its handling time), it is possible to either reduce or eliminate oscillations (which are present in the absence of additional food) in the ecosystem. On the other hand, if the system has stable coexistence initially, then it is possible to bring in oscillations and increase their amplitude till the target prey gets eradicated. Moreover, it is interesting to observe that it is possible to reduce and also eliminate the predator population by providing it with some additional food. This sounds contra-intuitive but theoretically possible, which depends on the choice of additional food, in particular its quantity and handling time. The section-wise split of this paper is as follows. In the next section, the model representing the dynamics of a predator–prey system in the presence of some additional food is introduced. In Section 3, the equilibrium analysis of the model, global dynamics of the considered ecosystem and its controllability are presented. In Section 4, the consequences of providing additional food to the predator on the predator–prey dynamics are discussed. Section 5 illustrates some of the key findings through numerical simulations followed by discussion and conclusions in Section 6. 2. The model Let us consider the following predator–prey model, given by N cNP N_ ¼ rN 1 , (1) K aþN bNP P_ ¼ mP, aþN
(2)
where r and K, respectively, represent the intrinsic growth rate and carrying capacity of the prey. m is the death rate of the predator in the absence of prey which is also termed as starvation rate. If h1 and e1 are two constants representing handling time of the predator per prey item and ability of the predator to detect the prey then we have c and a, representing the maximum rate of predation and half saturation value of the predator, to be 1=h1 and 1=e1 h1 , respectively. If e represents the efficiency with which the food consumed by the predator gets converted into predator biomass then b, the maximum growth rate of the predator is given by e=h1 . The system (1), (2) is well studied in the literature and analyzed from various perspectives (Kot, 2001; Rosenzweig, 1971; Srinivasu et al., 2001). We know that stability of the interior ¯ PÞ ¯ of the above system depends on its equilibrium ðN; position relative to the hump of the prey isocline. If it lies to the left (right) of this hump, then the equilibrium is unstable (asymptotically stable). Also, the above system admits a stable limit cycle about its unstable
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equilibrium which increases in amplitude as the equilibrium moves towards the P-axis from the hump. This occurrence of limit cycles at low equilibrium levels of the prey population is due to the instability nature of the interior equilibrium. Now, let us assume that the predator is provided with additional food of biomass A which is assumed to be distributed uniformly in the habitat as is the case with the prey as well as the predator in the habitat. We assume that the number of encounters per predator with the additional food is proportional to the density of the additional food. Here, the proportionality constant characterizes the ability of the predator to identify the additional food. Thus we have the following coupled differential system representing predator–prey dynamics when the predator is provided with additional food. N cNP N_ ¼ rN 1 , (3) K a þ aZA þ N bðN þ ZAÞP P_ ¼ mP. a þ aZA þ N
(4)
If h2 and e2 , respectively, represent the handling time of the predator per unit quantity of additional food and ability for the predator to detect the additional food, then we have Z ¼ e2 =e1 and a ¼ h2 =h1 . Here the term ZA represents effectual additional food level. Clearly the above system reduces to (1), (2) if A is taken to be zero. Now, we shall analyze the dynamics of the system (3), (4) and study its controllability with respect to the quantity and quality of the additional food. To reduce the number of parameters, we non-dimensionalize the systems (1)–(4) using the transformations x¼
N ; a
t0 ¼ rt;
y¼
cP ar
and obtain the following transformed systems, respectively. x xy bxy x_ ¼ x 1 ; y_ ¼ dy, (5) g 1þx 1þx x xy ; x_ ¼ x 1 g 1 þ ax þ x
y_ ¼
bðx þ xÞy dy, 1 þ ax þ x
(6)
where g¼
K ; a
b b¼ ; r
m ; r
x¼
f¯ ðxÞ ¼
x 1þx
d¼
ZA . a
(7)
Defining f ðxÞ ¼
x ; 1 þ ax þ x
(8)
and x x gðxÞ ¼ ð1 þ ax þ xÞ 1 ; g¯ ðxÞ ¼ ð1 þ xÞ 1 , g g
(9)
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the systems (5), (6), respectively, get reduced to x_ ¼ ½¯gðxÞ yf¯ ðxÞ; x_ ¼ ½gðxÞ yf ðxÞ;
y_ ¼ ½bf¯ ðxÞ dy, x d y. y_ ¼ bf ðxÞ 1 þ x
(10) (11)
3. Model analysis, global dynamics and controllability In this section, we analyze model (11) and draw conclusions regarding the asymptotic behavior of its trajectories. Observe that the prey isocline of (11) is an increasing function of both x and a in ½0; g which intersects the y-axis at ð0; 1 þ axÞ and x-axis at ðg; 0Þ. Thus, provision of additional food to the predator causes upward displacement to the prey isocline over the interval ½0; gÞ. The predator isocline of (11) is always a straight line just as that of (10) but it is a function of x and a. The predator isocline of (11) may move to the right or left from x ¼ d=ðb dÞ (the predator isocline of (10)) as x increases from zero, depending on the relative position of a with respect to b=d. The prey isocline of (11) attains its maximum at x ¼ ðg 1 axÞ=2 provided g41 þ ax whereas it is attained at x ¼ ðg 1Þ=240 for (10). This means that provision of additional food shifts the maximum of the prey isocline towards predator axis. Clearly, the systems (10) and (11) admit two common equilibrium points, viz., the trivial equilibrium ð0; 0Þ and axial equilibrium ðg; 0Þ while their interior equilibrium points are, respectively, represented by ðx; ¯ y¯ Þ and ðx ; y Þ, where ðx; ¯ y¯ Þ ¼ ðd=ðb dÞ; g¯ ðxÞÞ ¯ and ðx ; y Þ ¼ ððxðda bÞ þ dÞ=ðb dÞ; gðx ÞÞ (ref. Eq (9)). Clearly y 4¯y for x40. If aob=d ða4b=dÞ then x ox¯ ðx 4xÞ. ¯ Thus, it is important to note that, providing additional food may either decrease or increase the equilibrium level of the prey population depending on whether the ratio of handling times of additional food to prey item i.e., a is less than or greater than the ratio between the maximum birth rate to starvation rate ðb=dÞ, whereas it always enhances the equilibrium level of the predator when compared with that of (10). Also, for a4b=d, if (10) does not admit an interior equilibrium then (11) shall never admit interior equilibrium. If aob=d then the system (11) admits interior equilibrium even if (10) does not admit one provided x4ðd gðb dÞÞ=ðb daÞ and it will be relieved of this interior equilibrium if xXd=ðb daÞ. It is also important to observe that when (10) admits interior equilibrium point with a ¼ b=d, the equilibrium predator population increases with additional food but the equilibrium prey population remains at the same level as that of (10). In order to analyze the nature of the paths of the system (11), we require to evaluate the associated Jacobian at each of the above equilibrium points, which is given by 2 0 3 f ðxÞ g ðxÞf ðxÞ þ ½gðxÞ yf 0 ðxÞ 6 7 x f ðxÞx x J ðx;yÞ ¼ 4 . by f 0 ðxÞ 1 þ 2 d5 bf ðxÞ 1 þ x x x (12)
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Evaluating J at these points we obtain 2 3 1 0 5 bx J ð0;0Þ ¼ 4 0 d , 1 þ ax 2 6 J ðg;0Þ ¼ 6 4
1 0
ða; xÞ space:
3 g 1 þ ax þ g 7 7 5 gþx b d 1 þ ax þ g
and 2
J ðx ;y Þ
3 g0 ðx Þf ðx Þ f ðx Þ 6 7 x f ðx Þx ¼4 , bgðx Þ f 0 ðx Þ 1 þ 2 0 5 x x
where f 0 ðxÞ ¼
1 þ ax 40, ð1 þ ax þ xÞ2
1 ax 2x g ðxÞ ¼ 1 . g g g 0
(13)
(14)
Analyzing the eigen values of the Jacobians at the respective equilibrium point we derive information regarding the dynamics and eventual state of the system and their dependance on the system parameters. We treat b; d and g as ecosystem characteristic parameters while x and a as control parameters which characterize the quantity of additional food (made) available to the predator and its quality relative to the prey (with reference to their handling times), respectively. The global dynamics of the system (11) can be revealed under the following three natural conditions involving the existence and stability of the interior equilibrium point for the system (10). These conditions are given by d bþd o , bd bd
(15)
d bþd ogp , bd bd
(16)
d bþd o pg. bd bd
(17)
gp
It is relevant to note that the system (10) admits an interior equilibrium point if g4d=ðb dÞ, and this interior equilibrium is unstable (asymptotically stable) if g4ðoÞ ðb þ dÞ=ðb dÞ. Thus the interior equilibrium of (10) undergoes Hopf bifurcation at g ¼ ðb þ dÞ=ðb dÞ. From J ð0;0Þ , J ðg;0Þ and J ðx ;y Þ we infer that the nature of the equilibrium points ð0; 0Þ; ðg; 0Þ and ðx ; y Þ depends on the signs of the expressions bx=ð1 þ axÞ d; (an eigen value of J ð0;0Þ Þ; bððg þ xÞ=ð1 þ ax þ gÞÞ d (an eigen value of J ðg;0Þ Þ and g0 ðx Þ (the derivative of the prey isocline at x ). Hence, we consider the following curves (related to the expressions mentioned above) in the positive quadrant of
bx dð1 þ axÞ ¼ 0,
(18)
bðg þ xÞ dð1 þ ax þ gÞ ¼ 0,
(19)
2bx ðb þ dÞðax þ 1Þ þ ðb dÞg ¼ 0.
(20)
Note that each of these curves divides the positive quadrant into two regions which characterize the nature of associated equilibrium point. It is easy to observe that a ¼ b=d is an asymptote for both (18) and (19) while 2b=ðb þ dÞ is an asymptote for (20). Also the point of intersection between Eqs. (18) and (20) is ðbð1 1=gÞ=d; dg=bÞ. The dynamics of the system (11) is analyzed through the curves (18)–(20) which are presented in Figs. 1–3 under each of the conditions (15)–(17). From Fig. 1, observe that for a fixed a, as x increases from zero we may enter either D11 or D12 depending on the value of a. In either case the paths converge to ðg; 0Þ. In this region ð0; 0Þ is coexisting equilibrium with saddle nature. As we enter C 1 from D11 , the interior equilibrium ðx ; y Þ comes into existence with transcritical bifurcation taking place due to exchange of stability between ðg; 0Þ and ðx ; y Þ. Here, the nature of ð0; 0Þ remains unchanged. Moving from C 1 to B1 , ðx ; y Þ turns unstable retaining the saddle nature of ðg; 0Þ. Here there is Hopf bifurcation at the interior equilibrium. Moving into A1 , either from B1 or from C 1 (depending on the value of a i.e., 0oaobð1 1=gÞ=d or bð1 1=gÞ=doaob=d), the interior equilibrium vanishes leaving no change in the saddle nature of ðg; 0Þ and there is a saddle–node bifurcation at ð0; 0Þ. To understand Fig. 2, we divide it into three regions viz., 0oaobð1 1=gÞ=d (region i), bð1 1=gÞ=dpapb=d (region ii) and a4b=d (region iii). In region i, as we increase x from zero we enter C 21 where ðx ; y Þ exists and
Fig. 1. Eqs. (18), (20) and (19) define the boundaries between A1 ; B1 [ C 1 and B1 ; C 1 and C 1 ; D11 , respectively. The boundary between D11 and D12 is the line a ¼ b=d. In A1 ; ð0; 0Þ is unstable node and ðg; 0Þ is saddle. In B1 [ C 1 , ð0; 0Þ and ðg; 0Þ are saddles and the system admits an interior equilibrium which is unstable in B1 and asymptotically stable in C 1 . In D11 [ D12 , ð0; 0Þ is saddle and ðg; 0Þ is asymptotically stable.
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Fig. 2. Eqs. (18), (20) and (19) define the boundaries between A2 ; B2 [ C 21 and B2 ; C 21 and C 22 ; D2 . The boundary between C 21 and C 22 is the line a ¼ b=d. In A2 , ð0; 0Þ is unstable node and ðg; 0Þ is saddle. In B2 [ C 21 [ C 22 , ð0; 0Þ and ðg; 0Þ are saddles and interior equilibrium point exists. In region B2 , the interior equilibrium point is unstable and it is stable in the region C 21 [ C 22 . In D2 , ð0; 0Þ is saddle and ðg; 0Þ is asymptotically stable.
Fig. 3. Eqs. (18), (20) and (19) define the boundaries between A3 ; B31 [ C 31 and B31 [ B32 , C 31 [ C 32 and C 32 ; D3 . The equation of line which forms part of the boundary between C 31 and C 32 is a ¼ b=d. In A3 , ð0; 0Þ is unstable node and ðg; 0Þ is saddle. In B31 [ B32 [ C 31 [ C 32 , ð0; 0Þ and ðg; 0Þ are saddles and interior equilibrium point exists. In region B31 [ B32 , the interior equilibrium point is unstable and it is stable in the region C 31 [ C 32 . In D3 , ð0; 0Þ is saddle and ðg; 0Þ is asymptotically stable.
is globally asymptotically stable while ðg; 0Þ and ð0; 0Þ are saddles. From C 21 as we enter B2 there is Hopf bifurcation as ðx ; y Þ turns into unstable node or focus. As a result we have emergence of a unique limit cycle about ðx ; y Þ which is globally asymptotically stable.2 In this region both (0,0) 2
The existence of unique limit cycle, its global asymptotic stability and global asymptotic stability of stable interior equilibrium can be established by using the standard results (Glendinning, 1994).
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and ðg; 0Þ are saddles. Increasing x further, we move from B2 into A2 resulting in a saddle–node bifurcation at ð0; 0Þ and disappearance of ðx ; y Þ. ðg; 0Þ continues to be saddle in this region. In region ii, by increasing x we move from C 21 into A2 causing a saddle–node bifurcation at ð0; 0Þ and disappearance of the stable interior equilibrium point. ðg; 0Þ is always saddle in this region. In region iii, as we increase x we move into C 22 wherein the system admits all the three equilibrium points with ðx ; y Þ being globally asymptotically stable and the remaining two being saddles. As we move from C 22 into D2 the interior equilibrium point vanishes and it exchanges stability with ðg; 0Þ consequently there is a transcritical bifurcation at ðg; 0Þ. ð0; 0Þ is always saddle in region iii. In Fig. 3, we divide the parameter space into three regions given by 0oaobð1 1=gÞ=d (region i), bð1 1=gÞ=dpapb=d (region ii) and a4b=d (region iii). In region i, we note that as x increases from zero, we enter the region B31 where the system admits an un stable interior equilibrium ðx ; y Þ and two other equilibrium points ð0; 0Þ; ðg; 0Þ which are saddles in nature. Consequently, the system admits a unique stable limit cycle about ðx ; y Þ. Increasing x further, we move into A3 where the system admits only the trivial and axial equilibrium points with a saddle–node bifurcation at ð0; 0Þ while ðg; 0Þ remains saddle. In region ii, as x increases we move from B31 into C 31 with Hopf bifurcation at the interior equilibrium point. As a result the interior equilibrium becomes globally asymptotically stable. The remaining two equilibriums are saddles. The qualitative behavior of the system when in B31 is same as in that of region i. In region iii, as x increases we move through the regions B32 ; C 32 and D3 . The qualitative behavior of the system in B32 is identical with that of B31 . As we enter C 32 the interior equilibrium stabilizes due to Hopf bifurcation which is also globally asymptotically stable while the remaining equilibrium points are saddles. Moving from C 32 into D3 , transcritical bifurcation takes place at ðg; 0Þ due to exchange of stability between ðx ; y Þ and ðg; 0Þ. In D3 the system admits only the trivial and axial equilibriums with the later being globally asymptotically stable. Now, let us consider the case where a ¼ b=d. Here, if the system (10) does not admit an interior equilibrium then (11) will not admit any interior equilibrium for any x40 (Fig. 1). On the other hand, if (10) admits an interior equilibrium then (11) also will admit an interior equilibrium and its stability nature depends on the nature of ðx; ¯ y¯ Þ and food supply level x. If ðx; ¯ y¯ Þ is asymptotically stable then ðx ; y Þ is also globally asymptotically stable for any x40 (Fig. 2). But if ðx; ¯ y¯ Þ is unstable then ðx ; y Þ will be unstable for x 2 ½0; ½ðb þ dÞ ðb dÞgd=bðd bÞÞ (hence admits a unique globally asymptotically stable limit cycle) and it becomes globally asymptotically stable for x4½ðb þ dÞ ðb dÞgd=bðd bÞ. Thus, the interior equilibrium undergoes Hopf bifurcation at x ¼ ½ðb þ dÞ ðb dÞgd=bðd bÞ (Fig. 3).
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In each of the regions A1 , A2 and A3 (Figs. 1–3) the system (11) admits only the trivial equilibrium which is an unstable node and the axial equilibrium ðg; 0Þ which is a saddle. It is easy to observe that, when the control parameters belong to any of these regions, the stable manifold of ðg; 0Þ connects ð0; 0Þ and ðg; 0Þ along prey axis, while its unstable manifold separates the positive quadrant into two connected regions. This unstable manifold touches the y-axis and from there it continues to move on the y-axis heading towards þ1. Hence, those paths which lie below this unstable manifold eventually move towards the y-axis and reach it in a finite time under the influence of the unstable manifold and hence are likely to reach the y-axis at a lower level than that of the unstable manifold. This situation reverses for those paths initiating above the unstable manifold. Upon reaching the y-axis the solution moves on it in the increasing direction of y. Hence the system predicts unbounded growth for the predators when the control parameters belong to any of the regions A1 , A2 or A3 . Also note that, after the system becomes free from the prey, the predators can be maintained at any level by switching to the food supply x to d=ðb daÞ, after the predator population reaches the required level. Here, both the unlimited growth for the predators and the possibility to maintain the predators at any level with the same amount of food supply are somewhat unrealistic. This is due to our not assuming intra-specific competition among the predators at higher concentrations. From the above analysis, it can be observed that if the additional food supply level and its quality satisfies g 1 axo0
(21)
then the interior equilibrium of (11) is globally asymptotically stable. Otherwise, the stability of the interior equilibrium depends on the equilibrium prey population level. In fact, this analysis makes it possible for us to choose an appropriate additional food and its supply level so that the ecosystem can be driven to either a desired level or to a cycle about a specified equilibrium population. For, ~ yÞ ~ is the desired equilibrium state for the ecosystem, if ðx; then this will be admitted by the system (11) if a40 and x40 are chosen to satisfy ax þ 1 þ x~ ¼
~ b yg ¼ ðx~ þ xÞ. g x~ d
(22)
This equilibrium can be reached asymptotically if (21) is satisfied. On the other hand, if g 1 ax40
(23)
~ y~ can be reached asymptotically if then x; ~ x4
g 1 ax 2
(24)
else the state will reach a limit cycle asymptotically which is ~ yÞ. ~ centered around ðx;
4. Consequences of providing additional food In this section, we study the effect of providing the predator with additional food on the ecosystem dynamics. As observed in the previous section this effect is twofold. The system takes two opposite directions depending on the position of a relative to b=d. We characterize the additional food to be of high quality if aob=d and it is of low quality if a4b=d. Originally, if the ecosystem supports coexistence of prey and predators (i.e., the system (10) admits an interior equilibrium point whether stable or unstable) then continuous supply of high-quality additional food to the predators with the supply level x 2 ½0; d=ðb daÞÞ decreases the equilibrium prey population from d=ðb dÞ. Increasing the additional food supply beyond this interval eradicates the prey from the ecosystem in a finite time, and from that time the predators survive only on the external food supply. From this, we conclude that high-quality additional food source favors the biological control by benefiting the predator enough to suppress the target prey. On the other hand, providing predators with low-quality additional food the equilibrium prey–predator population moves from ðx ¼ d=ðb dÞ; gðx ÞÞ and heads towards ðg; 0Þ as the food supply x increases in ½0; ðgðb dÞ dÞ= ðda bÞÞ. Increasing the food supply beyond the above interval leads the prey population to its carrying capacity and eliminates the predator from the ecosystem. Thus lowquality additional source does not benefit the biological control. On the contrary, it benefits the prey as it reduces predation pressure up on the prey. In fact, if the added food is very dense and the handling time for the food item is high, predator individuals encounter the food effectively to the exclusion of the prey. However, because handling time for the added food is so large, the predator cannot consume food fast enough to equal or exceed the predator death rate. The predator therefore goes extinct and the prey moves to its equilibrium density. The theory predicts such a situation as the predators are not assumed to be optimal foragers. In the case where a ¼ b=d, if in the absence of additional food the system has stable coexistence then increasing the food supply increases the predator equilibrium level but does not change the equilibrium level of prey population or the nature of the interior equilibrium. On the other hand, if the system is oscillatory in the absence of additional food then these oscillations can be eliminated with higher level of food supply, i.e., x4½ðb þ dÞ ðb dÞgd=bðd bÞ. From the control parameter space analysis we observe that the additional food characteristics i.e., predator’s handling time for the additional food plays a very important role in the controllability of the ecosystem. With suitable choice on the additional food and its supply level the ecosystem can be made either prey dominated or predator dominated. The system can also be steered to a desired level subject to the parameter values satisfying required conditions. If a specific choice of food, supplied at
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a chosen level, drives the system oscillatory then we can eliminate such fluctuations by choosing a substitute for the additional food which stabilizes the system by eliminating the oscillations. Thus, through additional food we can eliminate either the predator alone and allow the prey to reach its carrying capacity or eliminate prey alone and subsequently maintain the predators in the ecosystem with the additional food. We can eliminate both prey and predator altogether by withdrawing the additional food after the prey gets eradicated from the ecosystem. 5. Numerical illustration In this section we illustrate some of the key findings using numerical simulation. We assume the values of the ecosystem parameters g, b and d to be 6.0, 0.3 and 0.2, respectively. Clearly these values satisfy (17) indicating existence of a unique asymptotically stable limit cycle for the system (10) about the equilibrium ð2:0; 2:0Þ. A path with initial value ð1; 1Þ approaching this limit cycle is shown in Fig. 4. Here we have b=d ¼ 1:5. In the simulations to follow we wish to illustrate the controllability of the system (11). ~ yÞ ~ to be (3.0, 3.0). From (22), Now, let us fix the target ðx; we have the values of a and x to be 2.0 (4b=d) and 1.0 which satisfy (24). Therefore, it is possible to reach the target by choosing the quality of additional food and its quantity as 2.0 and 1.0, respectively. Fig. 5A presents the path of the system (11) which initiates at ð1; 1Þ. This illustrates the possibility to eliminate cycles in the ecosystem through provision of additional food to predators. Observe that we are moving from the region B32 into C 32 in Fig. 3 with the above choices for a and x. Fixing the target population level to be (1.0, 3.0), we obtain the values of a and x to be 1.1429 (ob=d) and 1.4 (from (22)). As these parameter values do not satisfy (24),
Fig. 4. Numerical illustration showing predator and prey evolution in the absence of additional food.
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the state of the system cannot reach the target but tends to a limit cycle about the target. This illustrates the change in the dynamics due to moving upwards in the region B31 from a-axis in Fig. 3. Path of the system (11) initiating at ð1; 1Þ is presented in Fig. 5B. Observe that providing additional food, with the specifications mentioned above, compresses the limit cycle and moves it towards the y-axis. Suppose we wish to retain the prey population at 2:0 only but increase the predator population to 5:0. From the analysis presented in the previous sections, we have a ¼ b=d ¼ 1:5 and from Eq. (22) we obtain x ¼ 3:0. Since these parameter values satisfy (24), we have the target state ð2:0; 5:0Þ to be an asymptotically stable equilibrium. Fig. 5C presents path of the system (11) which illustrates the approach to the target state in the presence of additional food. If we fix the target level to be ð2:0; 2:45Þ, as earlier we have a ¼ 1:5 and from (22) we obtain x to be 0:45. With these parameter values we approach a limit cycle about ð2:0; 2:45Þ asymptotically as (24) is not satisfied by the control parameter values. Here, the parameter values lie on the line a ¼ b=d ¼ 1:5 with x ¼ 0:45 2 ð0; 23Þ ¼ ð0; ½ðb þ dÞ=ðb dÞgd=bðd bÞÞ. The corresponding simulation for the system (11) is shown in Fig. 5D. 6. Discussion and conclusions In spite of growing prominence in practical biological control as well as theoretical study for studying the consequences on predator–prey dynamics due to provision of additional foods to the predators, there is increasing need to bridge these two areas explicitly. To improve our understanding on these aspects, it is important to have a closer look at various mechanisms, assumptions and hypotheses involved in the theoretical models, built to represent predator–prey dynamics in the presence of additional food to predators. In this paper, we make a systematic analysis of the dynamics of a predator–prey system when some additional food is provided to the predator. The additional food is assumed to be either non-reproducing prey or some food source. We do not make any distinction regarding the additional food like complementary, substitutable, essential or alternative (Hodek and Honek, 1996; Sabelis and van Rijn, 2005). We only assume that the predator is capable of reproducing by consuming either of the available food, and we discuss about the food quality only when relevant to understanding the model or predictions. We characterize quality of a food with respect to its handling time. The dynamics of the predator–prey system in the presence of additional food is viewed as a variation of predator–prey model with Holling type II functional response. From the analysis of the model, we observe that the ratio of predator handling times for the additional food and prey plays a key role in the controllability of the ecosystem. We have three cases here. In the first case we have the ratio of handling times mentioned above is greater than the ratio between the
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Fig. 5. Numerical illustrations showing predator and prey evolution in the presence of additional food with different handling times and supply levels.
maximum predator growth rate and its starvation rate. We have the above relation reversed in case 2. In case 3, we have the ratios to be equal. In the first case, we clearly have the handling time for the additional food to be higher than that of prey. Here, we find that, if the predators go extinct in the absence of additional food, it is not possible to bring in eventual predator–prey coexistence by providing additional food to predators. Therefore, if in the absence of additional food, the births due to consumption of prey are not compensating the deaths, then adding any amount of additional food cannot improve the situation. This is due to the distraction effect caused by the addition of low-quality food which takes more time to consume when compared with that of a prey item and the predators are time limited. Thus, in this situation biological control is not possible through additional food. Instead, if the predators and prey have a stable coexistence in the absence of additional food, then the
distraction caused by the provision of additional food to the predators partially relieves the prey from predation pressure, and also decreases the birth rate of the predator due to higher value of the handling time for the additional food. As the predators encounter the additional food they spend more time in consuming the additional food and a balancing quantity of the prey is released. This observation goes with the inferences made by Harwood and Obrycki (2005), wherein it was concluded that—Although conservation biological control may enhance predator growth rate by providing an abundant and nutritionally balanced diet, it is feasible that predators will divert feeding efforts towards non-pest food items, thus reducing biological control. Also, increase in the additional food (greater than a critical value) decreases the birth rate of the predator. Hence increase in the concentration of such additional food increases the distraction effect causing a decline in the predator population and increasing the prey population. Thus the coexistence continues as long as the additional
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food level is less than a critical value beyond which the net births due to consuming the available food does no longer compensate the deaths resulting in elimination of predators, and the prey reach their carrying capacity. In this case, if the predator–prey system is eventually oscillatory in the absence of additional food, then the increase in the prey population due to additional food reduces the amplitude of the oscillations and beyond a concentration level the ecosystem will enter stable coexistence. The effects due to further increase of additional food level from here will be as discussed above. For the second case, where the ratio between the handling times of the additional food and the prey is less than the ratio between the maximum predator growth rate and its starvation rate, several subcases are possible. If the predator goes extinct in the system without additional food, then we can bring in coexistence of predator and prey by providing the predator with additional food with concentration level belonging to a specified interval. Here, the additional food increases the consumption of the predator population which reflects in the numerical abundance of the predator which in turn increases the per capita consumption of prey. As a result, there is a decrease in the equilibrium prey population. If, in the absence of additional food, the system has stable coexistence then providing additional food may either cause monotonic increase in the eventual value of the predator while maintaining stability, or it may bring in oscillations into the system when the supply goes beyond a specified level, leading to a stable limit cycle. These cycles move towards the predator axis with increase in the additional food supply. Beyond a certain level of food supply, the prey population goes extinct and predators are solely supported by the additional food. Even in the case where the eventual predator population is monotone with respect to food supply, the prey gets eradicated beyond a specific level of food supply. At this stage the predators can also be eliminated by with-drawing the additional food supply to the predators. If originally the predator and prey coexist in the absence of additional food but are oscillatory, then supplying the predators with additional food with sufficiently low handling time draws these oscillations closer to the predator axis in such way that beyond some supply level the prey population crashes and only predators survive eventually. Also it is possible to compress the oscillations and even eliminate them leading to their coexistence by providing food with relatively higher handling time. Whatever be the initial alterations in the behavior of the population due to increase in the additional food, increasing the food supply beyond a value always eradicates the prey from the ecosystem. It is important to note that even if the prey is of poor quality when compared to the additional food, the prey can be controlled biologically. This supports the inferences in Harwood and Obrycki (2005)—Simply because pests are a poor quality prey item (Toft, 2005) does not necessarily
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translate to little or no biological control in the field where generalist predators are frequently in a state of hunger e.g., (Bilde and Toft, 1998) and readily consume prey (Harwood et al., 2004, 2005). On the contrary, if the additional food happens to be of poor quality than prey, even then the prey population can be biologically controlled as long as it satisfies this second case. In the third case, if in the absence of additional food, the system admits an interior equilibrium, then introduction of additional food leaves the prey equilibrium level unaltered and the predator equilibrium increases with food supply. The nature of the equilibrium depends on the nature of the interior equilibrium in the absence of additional food. If it is asymptotically stable when there is no food supply, then it will continue to be asymptotically stable with increase in food supply. On the other hand, if it is unstable when there is no food supply, then the amplitude of the oscillations due to instability of the interior equilibrium decreases with increase in the food level and after a critical food level, the equilibrium turns asymptotically stable and it will remain in that state for all higher food levels. Finally, we conclude that the modified version of predator–prey model with Holling type II functional response, which represents predator–prey dynamics in the presence of additional food exhibits very interesting dynamics. Theoretically it is possible to control the dynamics of the system by manipulating the quality of the additional food and its supply level. It is possible to drive the prey and predator population to a desired level within specified limits. Also, it is possible to control and eliminate any oscillations present in the ecosystem by an appropriate choice of additional food and supply level. Oscillations can also be brought into the ecosystem through provision of additional food, if they are not present prior to the supply of food. We can not only eliminate prey by enhancing the predation effect by supplying the right kind of additional food to the predator, but also eliminate the predator by distracting the predator with a supply of low-quality additional food at high density which decreases the per capita growth rate of the predator below its starvation rate and releases the prey from predation pressure. In the context of biological pest control, the results caution the manager on the choice of quality and quantity of the additional food used for this purpose. An arbitrary choice may have completely opposite effects leading to increase in pest concentration and eradication of the predator. The analysis suggests a necessity to incorporate intra-specific competition for the predators at higher population level to make it more realistic.
Acknowledgments The authors are thankful to the three reviewers for their extremely valuable suggestions.
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References Bilde, T., Toft, S., 1998. Quantifying food limitation of arthropod predators in the field. Oecologia 115, 54–58. Coll, M., Guershon, 2002. Omnivoryin terrestrial arthropods: mixing plant and prey diets. Annu. Rev. Entomol. 47, 267–297. Glendinning, P., 1994. Stability, Instability and Chaos: an introduction to the theory of nonlinear differential equations. Cambridge Texts in Applied Mathematics. Cambridge University Press, USA. Harmon, J.P., 2003. Indirect interactions among a generalist predator and its multiple foods. Ph.D. Thesis, St. Paul, MN, University of Minnesota. Harwood, J.D., Obrycki, J.J., 2005. The role of alternative prey in sustaining predator populations. In: Hoddle, M.S. (Ed.), Proceedings of Second International Symposium on Biological Control of Arthropods, vol. II, pp. 453–462. Harwood, J.D., Sunderland, K.D., Symondson, W.O.C., 2004. Prey selection by linyphiid spiders: molecular tracking of the effects of alternative prey on rates of aphid consumption in the field. Mol. Ecology 13, 3549–3560. Harwood, J.D., Sunderland, K.D., Symondson, W.O.C., 2005. Monoclonal antibodies reveal the potential of the tetragnathids spider Pachygnatha degeeri (Araneae: Tetragnathidae) as an aphid predator. Bull. Entomol. Res. 95, 161–167. Hodek, I., Honek, A., 1996. Ecology of Coccinellide. Kluwer Academic Publishers, Dordrecht, The Netherlands. Holt, R.D., 1977. Predation, apparent competition, and the structure of prey communities. Theor. Popul. Biol. 12, 197–229.
Holt, R.D., 1984. Spatial heterogeneity, indirect interactions, and the coexistence of prey species. Am. Nat. 124, 377–406. Holt, R.D., Lawton, J.H., 1994. The ecological consequences of shared natural enemies. Annu. Rev. Ecol. Syst. 25, 495–520. Kot, M., 2001. Elements of Mathematical Ecology. Cambridge University Press, UK. Murdoch, W.W., Chesson, J., Chesson, P.L., 1985. Biological control in theory and practice. Am. Nat. 125 (3), 344–366. Rosenzweig, M.L., 1971. Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science 171, 385–387. Sabelis, M.W., van Rijn, P.C.J., 2005. When does alternative food promote biological pest control? In: Hoddle, M.S. (Ed.), Proceedings of Second International Symposium on Biological Control of Arthropods, vol. II, pp. 428–437. Srinivasu, P.D.N., Ismail, S., Naidu, Ch.R., 2001. Global dynamics and controllability of a harvested prey–predator system. J. Biol. Syst. 9 (1), 67–79. Toft, S., 2005. The quality of aphids as food for generalist predators: implications for natural control of aphids. Eur. J. Entomol. 102 (3), 371–383. van Baalen, M., Krˇ ivan, V., van Rijn, P.C.J., Sabelis, M.W., 2001. Alternative food, switching predators, and the persistence of predator–prey systems. Am. Nat. 157 (5), 512–524. van Rijn, P.C.J., van Houten, Y.M., Sabelis, M.W., 2002. How plants benefit from providing food to predators even when it is also edible to herbivores. Ecology 83, 2664–2679. Wootton, J.T., 1994. The nature and consequences of indirect effects in ecological communities. Annu. Rev. Ecol. Syst. 25, 443–466.
Theoretical Population Biology 72 (2007) 111–120 www.elsevier.com/locate/tpb
Biological control through provision of additional food to predators: A theoretical study P.D.N. Srinivasua,, B.S.R.V. Prasada, M. Venkatesulub,1 a
Department of Mathematics, Andhra University, Visakhapatnam 530 003, India Department of Mathematics & Statistics, College of Arts & Science, University of Missouri, Kansas City, Missouri-64110, USA
b
Received 29 September 2006 Available online 10 April 2007
Abstract In this paper we analyze a variation of a standard predator–prey model with type II functional response which represents predator–prey dynamics in the presence of some additional food to the predator. The aim is to study the consequences of providing additional food on the system dynamics. We conclude that handling times for the available foods play a key role in determining the eventual state of the ecosystem. It is interesting to observe that by varying the quality and quantity of additional food we can not only control and limit the prey, but also limit and eradicate the predators. In the context of biological pest control, the results caution the manager on the choice of quality and quantity of the additional food used for this purpose. An arbitrary choice may have opposite effects leading to increase in pest concentration and eradication of the predator. This study offers insight into the possible management strategies that involve manipulation of quality and supply level of additional food to predators, for the benefit of biological control. The theoretical conclusions agree with results of some practical biological control experiments. r 2007 Elsevier Inc. All rights reserved. Keywords: Biological control; Additional food; Predator; Prey; Pest; Handling time
1. Introduction The consequences of providing a predator with additional food and the corresponding effects on the predator–prey dynamics and its utility in biological control have been the topic of study for many scientists. A major portion of the literature, dealing with biological control aspects, assumes the role of pest for the prey. These studies address the controllability and eradication of the pest through the predator by providing the latter with alternative or additional food. This is modeled mathematically as one predator–two (non-interacting) prey system. Such shared predator models predict that adding additional prey (different from the one existing in the ecosystem Corresponding author. Fax: +91 891 2755324.
E-mail addresses: [email protected] (P.D.N. Srinivasu), [email protected] (B.S.R.V. Prasad), [email protected] (M. Venkatesulu). 1 Permanent address: Professor of Mathematics, Department of Computer Applications, Arulmigu Kalasalingam College of Engineering, Krishnankoil 626190, Tamil Nadu, India. 0040-5809/$ - see front matter r 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.tpb.2007.03.011
which could also be a non-reproducing prey or food source) increases predators and also their effect on target prey, thereby helping to decrease the abundance of prey (Holt and Lawton, 1994; van Baalen et al., 2001; van Rijn et al., 2002). This phenomenon, i.e., the presence of alternative food for a predator leading to reduction in equilibrium prey densities, is termed as apparent competition (Holt, 1977). A lot of theoretical work has been done in assessing the effects that can occur among the prey that share a predator. Some of these works can be found in the papers reviewed in Holt (1984) and also in the recent works of Harmon (2003). Most of these theoretical models conclude that adding a non-pest alternative prey to a predator–prey system would lower the density of the target prey. By and large, these works infer that manipulating non-target prey can influence the predator in a way that increases target predation and eventually controls the prey population. However, from empirical data it is clear that addition of alternative prey does not always increase target predation (Harwood and Obrycki, 2005; Holt and Lawton, 1994;
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Murdoch et al., 1985; Wootton, 1994). Thus, there is an apparent conflict between theory and empirical observations. In the recent works of Sabelis and van Rijn (2005), it was pointed out that alternative food need not always promote biological pest control and it depends on the role of this food, whether it is substitutable or complementary to the prey. As per the findings in Harwood and Obrycki (2005), the effect of consuming non-pest species on rates of pest predation by a generalist predator can be twofold; feeding upon these nutritious food items generally enhances fecundity thus improving their population growth, but the presence of alternative prey, especially during times when pest regulation is required, can result in reduced levels of pest consumption per individual predator. From the above observations we recognize that provision of additional food to a predator may or may not reduce the target prey density. This calls for a better understanding of the mechanisms, assumptions and biological hypothesis of predator–prey dynamic models involving additional food. Investigation in this direction is likely to improve our understanding on the consequences of providing additional food to a predator, thereby bringing out an explicit link between practical biological control and theoretical studies. In this paper, we consider a modified version of a predator–prey model with Holling type II predation to reflect the dynamics of predator and prey in the presence of some additional food for the predators. It is assumed that the additional food is not dynamic but maintained at a specific constant level either by the nature or by an external agency. The advantage of this approach is that it reduces the dimension of the system from three to two, which allows the phase plane analysis for studying the consequences of availability of additional food to the predator (van Baalen et al., 2001). This also describes ecosystem dynamics when a non-reproducing prey or food source is added to it (Holt and Lawton, 1994; van Baalen et al., 2001; van Rijn et al., 2002). In the model under consideration, it is assumed that the number of encounters per predator with the additional food is proportional to the density of additional food in the environment and the predators are not assumed to be optimal foragers. Here, we analyze the dynamics of this model in a systematic manner, study the dependance of the dynamics on some identified vital parameters and discuss the global behavior and controllability of the ecosystem. Our interest is to investigate the consequences of providing additional food to the predators on the predator–prey dynamics. From this study, it is possible to develop management strategies that manipulate quality of the additional food and its supply level for the benefit of biological control. This study illustrates that exploring the fundamental nature and ecological mechanisms of predator–food interactions, we will be in a position to understand the indirect interactions taking place in the ecosystem and their vital role in influencing the predators, which helps to predict and control the eventual state of the ecosystem and
ultimately determine prey–predator regulation (Coll and Guershon, 2002; Holt and Lawton, 1994; Wootton, 1994). In this study it is found that, by providing the predator with appropriate additional food (characterized by its handling time), it is possible to either reduce or eliminate oscillations (which are present in the absence of additional food) in the ecosystem. On the other hand, if the system has stable coexistence initially, then it is possible to bring in oscillations and increase their amplitude till the target prey gets eradicated. Moreover, it is interesting to observe that it is possible to reduce and also eliminate the predator population by providing it with some additional food. This sounds contra-intuitive but theoretically possible, which depends on the choice of additional food, in particular its quantity and handling time. The section-wise split of this paper is as follows. In the next section, the model representing the dynamics of a predator–prey system in the presence of some additional food is introduced. In Section 3, the equilibrium analysis of the model, global dynamics of the considered ecosystem and its controllability are presented. In Section 4, the consequences of providing additional food to the predator on the predator–prey dynamics are discussed. Section 5 illustrates some of the key findings through numerical simulations followed by discussion and conclusions in Section 6. 2. The model Let us consider the following predator–prey model, given by N cNP N_ ¼ rN 1 , (1) K aþN bNP P_ ¼ mP, aþN
(2)
where r and K, respectively, represent the intrinsic growth rate and carrying capacity of the prey. m is the death rate of the predator in the absence of prey which is also termed as starvation rate. If h1 and e1 are two constants representing handling time of the predator per prey item and ability of the predator to detect the prey then we have c and a, representing the maximum rate of predation and half saturation value of the predator, to be 1=h1 and 1=e1 h1 , respectively. If e represents the efficiency with which the food consumed by the predator gets converted into predator biomass then b, the maximum growth rate of the predator is given by e=h1 . The system (1), (2) is well studied in the literature and analyzed from various perspectives (Kot, 2001; Rosenzweig, 1971; Srinivasu et al., 2001). We know that stability of the interior ¯ PÞ ¯ of the above system depends on its equilibrium ðN; position relative to the hump of the prey isocline. If it lies to the left (right) of this hump, then the equilibrium is unstable (asymptotically stable). Also, the above system admits a stable limit cycle about its unstable
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equilibrium which increases in amplitude as the equilibrium moves towards the P-axis from the hump. This occurrence of limit cycles at low equilibrium levels of the prey population is due to the instability nature of the interior equilibrium. Now, let us assume that the predator is provided with additional food of biomass A which is assumed to be distributed uniformly in the habitat as is the case with the prey as well as the predator in the habitat. We assume that the number of encounters per predator with the additional food is proportional to the density of the additional food. Here, the proportionality constant characterizes the ability of the predator to identify the additional food. Thus we have the following coupled differential system representing predator–prey dynamics when the predator is provided with additional food. N cNP N_ ¼ rN 1 , (3) K a þ aZA þ N bðN þ ZAÞP P_ ¼ mP. a þ aZA þ N
(4)
If h2 and e2 , respectively, represent the handling time of the predator per unit quantity of additional food and ability for the predator to detect the additional food, then we have Z ¼ e2 =e1 and a ¼ h2 =h1 . Here the term ZA represents effectual additional food level. Clearly the above system reduces to (1), (2) if A is taken to be zero. Now, we shall analyze the dynamics of the system (3), (4) and study its controllability with respect to the quantity and quality of the additional food. To reduce the number of parameters, we non-dimensionalize the systems (1)–(4) using the transformations x¼
N ; a
t0 ¼ rt;
y¼
cP ar
and obtain the following transformed systems, respectively. x xy bxy x_ ¼ x 1 ; y_ ¼ dy, (5) g 1þx 1þx x xy ; x_ ¼ x 1 g 1 þ ax þ x
y_ ¼
bðx þ xÞy dy, 1 þ ax þ x
(6)
where g¼
K ; a
b b¼ ; r
m ; r
x¼
f¯ ðxÞ ¼
x 1þx
d¼
ZA . a
(7)
Defining f ðxÞ ¼
x ; 1 þ ax þ x
(8)
and x x gðxÞ ¼ ð1 þ ax þ xÞ 1 ; g¯ ðxÞ ¼ ð1 þ xÞ 1 , g g
(9)
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the systems (5), (6), respectively, get reduced to x_ ¼ ½¯gðxÞ yf¯ ðxÞ; x_ ¼ ½gðxÞ yf ðxÞ;
y_ ¼ ½bf¯ ðxÞ dy, x d y. y_ ¼ bf ðxÞ 1 þ x
(10) (11)
3. Model analysis, global dynamics and controllability In this section, we analyze model (11) and draw conclusions regarding the asymptotic behavior of its trajectories. Observe that the prey isocline of (11) is an increasing function of both x and a in ½0; g which intersects the y-axis at ð0; 1 þ axÞ and x-axis at ðg; 0Þ. Thus, provision of additional food to the predator causes upward displacement to the prey isocline over the interval ½0; gÞ. The predator isocline of (11) is always a straight line just as that of (10) but it is a function of x and a. The predator isocline of (11) may move to the right or left from x ¼ d=ðb dÞ (the predator isocline of (10)) as x increases from zero, depending on the relative position of a with respect to b=d. The prey isocline of (11) attains its maximum at x ¼ ðg 1 axÞ=2 provided g41 þ ax whereas it is attained at x ¼ ðg 1Þ=240 for (10). This means that provision of additional food shifts the maximum of the prey isocline towards predator axis. Clearly, the systems (10) and (11) admit two common equilibrium points, viz., the trivial equilibrium ð0; 0Þ and axial equilibrium ðg; 0Þ while their interior equilibrium points are, respectively, represented by ðx; ¯ y¯ Þ and ðx ; y Þ, where ðx; ¯ y¯ Þ ¼ ðd=ðb dÞ; g¯ ðxÞÞ ¯ and ðx ; y Þ ¼ ððxðda bÞ þ dÞ=ðb dÞ; gðx ÞÞ (ref. Eq (9)). Clearly y 4¯y for x40. If aob=d ða4b=dÞ then x ox¯ ðx 4xÞ. ¯ Thus, it is important to note that, providing additional food may either decrease or increase the equilibrium level of the prey population depending on whether the ratio of handling times of additional food to prey item i.e., a is less than or greater than the ratio between the maximum birth rate to starvation rate ðb=dÞ, whereas it always enhances the equilibrium level of the predator when compared with that of (10). Also, for a4b=d, if (10) does not admit an interior equilibrium then (11) shall never admit interior equilibrium. If aob=d then the system (11) admits interior equilibrium even if (10) does not admit one provided x4ðd gðb dÞÞ=ðb daÞ and it will be relieved of this interior equilibrium if xXd=ðb daÞ. It is also important to observe that when (10) admits interior equilibrium point with a ¼ b=d, the equilibrium predator population increases with additional food but the equilibrium prey population remains at the same level as that of (10). In order to analyze the nature of the paths of the system (11), we require to evaluate the associated Jacobian at each of the above equilibrium points, which is given by 2 0 3 f ðxÞ g ðxÞf ðxÞ þ ½gðxÞ yf 0 ðxÞ 6 7 x f ðxÞx x J ðx;yÞ ¼ 4 . by f 0 ðxÞ 1 þ 2 d5 bf ðxÞ 1 þ x x x (12)
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Evaluating J at these points we obtain 2 3 1 0 5 bx J ð0;0Þ ¼ 4 0 d , 1 þ ax 2 6 J ðg;0Þ ¼ 6 4
1 0
ða; xÞ space:
3 g 1 þ ax þ g 7 7 5 gþx b d 1 þ ax þ g
and 2
J ðx ;y Þ
3 g0 ðx Þf ðx Þ f ðx Þ 6 7 x f ðx Þx ¼4 , bgðx Þ f 0 ðx Þ 1 þ 2 0 5 x x
where f 0 ðxÞ ¼
1 þ ax 40, ð1 þ ax þ xÞ2
1 ax 2x g ðxÞ ¼ 1 . g g g 0
(13)
(14)
Analyzing the eigen values of the Jacobians at the respective equilibrium point we derive information regarding the dynamics and eventual state of the system and their dependance on the system parameters. We treat b; d and g as ecosystem characteristic parameters while x and a as control parameters which characterize the quantity of additional food (made) available to the predator and its quality relative to the prey (with reference to their handling times), respectively. The global dynamics of the system (11) can be revealed under the following three natural conditions involving the existence and stability of the interior equilibrium point for the system (10). These conditions are given by d bþd o , bd bd
(15)
d bþd ogp , bd bd
(16)
d bþd o pg. bd bd
(17)
gp
It is relevant to note that the system (10) admits an interior equilibrium point if g4d=ðb dÞ, and this interior equilibrium is unstable (asymptotically stable) if g4ðoÞ ðb þ dÞ=ðb dÞ. Thus the interior equilibrium of (10) undergoes Hopf bifurcation at g ¼ ðb þ dÞ=ðb dÞ. From J ð0;0Þ , J ðg;0Þ and J ðx ;y Þ we infer that the nature of the equilibrium points ð0; 0Þ; ðg; 0Þ and ðx ; y Þ depends on the signs of the expressions bx=ð1 þ axÞ d; (an eigen value of J ð0;0Þ Þ; bððg þ xÞ=ð1 þ ax þ gÞÞ d (an eigen value of J ðg;0Þ Þ and g0 ðx Þ (the derivative of the prey isocline at x ). Hence, we consider the following curves (related to the expressions mentioned above) in the positive quadrant of
bx dð1 þ axÞ ¼ 0,
(18)
bðg þ xÞ dð1 þ ax þ gÞ ¼ 0,
(19)
2bx ðb þ dÞðax þ 1Þ þ ðb dÞg ¼ 0.
(20)
Note that each of these curves divides the positive quadrant into two regions which characterize the nature of associated equilibrium point. It is easy to observe that a ¼ b=d is an asymptote for both (18) and (19) while 2b=ðb þ dÞ is an asymptote for (20). Also the point of intersection between Eqs. (18) and (20) is ðbð1 1=gÞ=d; dg=bÞ. The dynamics of the system (11) is analyzed through the curves (18)–(20) which are presented in Figs. 1–3 under each of the conditions (15)–(17). From Fig. 1, observe that for a fixed a, as x increases from zero we may enter either D11 or D12 depending on the value of a. In either case the paths converge to ðg; 0Þ. In this region ð0; 0Þ is coexisting equilibrium with saddle nature. As we enter C 1 from D11 , the interior equilibrium ðx ; y Þ comes into existence with transcritical bifurcation taking place due to exchange of stability between ðg; 0Þ and ðx ; y Þ. Here, the nature of ð0; 0Þ remains unchanged. Moving from C 1 to B1 , ðx ; y Þ turns unstable retaining the saddle nature of ðg; 0Þ. Here there is Hopf bifurcation at the interior equilibrium. Moving into A1 , either from B1 or from C 1 (depending on the value of a i.e., 0oaobð1 1=gÞ=d or bð1 1=gÞ=doaob=d), the interior equilibrium vanishes leaving no change in the saddle nature of ðg; 0Þ and there is a saddle–node bifurcation at ð0; 0Þ. To understand Fig. 2, we divide it into three regions viz., 0oaobð1 1=gÞ=d (region i), bð1 1=gÞ=dpapb=d (region ii) and a4b=d (region iii). In region i, as we increase x from zero we enter C 21 where ðx ; y Þ exists and
Fig. 1. Eqs. (18), (20) and (19) define the boundaries between A1 ; B1 [ C 1 and B1 ; C 1 and C 1 ; D11 , respectively. The boundary between D11 and D12 is the line a ¼ b=d. In A1 ; ð0; 0Þ is unstable node and ðg; 0Þ is saddle. In B1 [ C 1 , ð0; 0Þ and ðg; 0Þ are saddles and the system admits an interior equilibrium which is unstable in B1 and asymptotically stable in C 1 . In D11 [ D12 , ð0; 0Þ is saddle and ðg; 0Þ is asymptotically stable.
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Fig. 2. Eqs. (18), (20) and (19) define the boundaries between A2 ; B2 [ C 21 and B2 ; C 21 and C 22 ; D2 . The boundary between C 21 and C 22 is the line a ¼ b=d. In A2 , ð0; 0Þ is unstable node and ðg; 0Þ is saddle. In B2 [ C 21 [ C 22 , ð0; 0Þ and ðg; 0Þ are saddles and interior equilibrium point exists. In region B2 , the interior equilibrium point is unstable and it is stable in the region C 21 [ C 22 . In D2 , ð0; 0Þ is saddle and ðg; 0Þ is asymptotically stable.
Fig. 3. Eqs. (18), (20) and (19) define the boundaries between A3 ; B31 [ C 31 and B31 [ B32 , C 31 [ C 32 and C 32 ; D3 . The equation of line which forms part of the boundary between C 31 and C 32 is a ¼ b=d. In A3 , ð0; 0Þ is unstable node and ðg; 0Þ is saddle. In B31 [ B32 [ C 31 [ C 32 , ð0; 0Þ and ðg; 0Þ are saddles and interior equilibrium point exists. In region B31 [ B32 , the interior equilibrium point is unstable and it is stable in the region C 31 [ C 32 . In D3 , ð0; 0Þ is saddle and ðg; 0Þ is asymptotically stable.
is globally asymptotically stable while ðg; 0Þ and ð0; 0Þ are saddles. From C 21 as we enter B2 there is Hopf bifurcation as ðx ; y Þ turns into unstable node or focus. As a result we have emergence of a unique limit cycle about ðx ; y Þ which is globally asymptotically stable.2 In this region both (0,0) 2
The existence of unique limit cycle, its global asymptotic stability and global asymptotic stability of stable interior equilibrium can be established by using the standard results (Glendinning, 1994).
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and ðg; 0Þ are saddles. Increasing x further, we move from B2 into A2 resulting in a saddle–node bifurcation at ð0; 0Þ and disappearance of ðx ; y Þ. ðg; 0Þ continues to be saddle in this region. In region ii, by increasing x we move from C 21 into A2 causing a saddle–node bifurcation at ð0; 0Þ and disappearance of the stable interior equilibrium point. ðg; 0Þ is always saddle in this region. In region iii, as we increase x we move into C 22 wherein the system admits all the three equilibrium points with ðx ; y Þ being globally asymptotically stable and the remaining two being saddles. As we move from C 22 into D2 the interior equilibrium point vanishes and it exchanges stability with ðg; 0Þ consequently there is a transcritical bifurcation at ðg; 0Þ. ð0; 0Þ is always saddle in region iii. In Fig. 3, we divide the parameter space into three regions given by 0oaobð1 1=gÞ=d (region i), bð1 1=gÞ=dpapb=d (region ii) and a4b=d (region iii). In region i, we note that as x increases from zero, we enter the region B31 where the system admits an un stable interior equilibrium ðx ; y Þ and two other equilibrium points ð0; 0Þ; ðg; 0Þ which are saddles in nature. Consequently, the system admits a unique stable limit cycle about ðx ; y Þ. Increasing x further, we move into A3 where the system admits only the trivial and axial equilibrium points with a saddle–node bifurcation at ð0; 0Þ while ðg; 0Þ remains saddle. In region ii, as x increases we move from B31 into C 31 with Hopf bifurcation at the interior equilibrium point. As a result the interior equilibrium becomes globally asymptotically stable. The remaining two equilibriums are saddles. The qualitative behavior of the system when in B31 is same as in that of region i. In region iii, as x increases we move through the regions B32 ; C 32 and D3 . The qualitative behavior of the system in B32 is identical with that of B31 . As we enter C 32 the interior equilibrium stabilizes due to Hopf bifurcation which is also globally asymptotically stable while the remaining equilibrium points are saddles. Moving from C 32 into D3 , transcritical bifurcation takes place at ðg; 0Þ due to exchange of stability between ðx ; y Þ and ðg; 0Þ. In D3 the system admits only the trivial and axial equilibriums with the later being globally asymptotically stable. Now, let us consider the case where a ¼ b=d. Here, if the system (10) does not admit an interior equilibrium then (11) will not admit any interior equilibrium for any x40 (Fig. 1). On the other hand, if (10) admits an interior equilibrium then (11) also will admit an interior equilibrium and its stability nature depends on the nature of ðx; ¯ y¯ Þ and food supply level x. If ðx; ¯ y¯ Þ is asymptotically stable then ðx ; y Þ is also globally asymptotically stable for any x40 (Fig. 2). But if ðx; ¯ y¯ Þ is unstable then ðx ; y Þ will be unstable for x 2 ½0; ½ðb þ dÞ ðb dÞgd=bðd bÞÞ (hence admits a unique globally asymptotically stable limit cycle) and it becomes globally asymptotically stable for x4½ðb þ dÞ ðb dÞgd=bðd bÞ. Thus, the interior equilibrium undergoes Hopf bifurcation at x ¼ ½ðb þ dÞ ðb dÞgd=bðd bÞ (Fig. 3).
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In each of the regions A1 , A2 and A3 (Figs. 1–3) the system (11) admits only the trivial equilibrium which is an unstable node and the axial equilibrium ðg; 0Þ which is a saddle. It is easy to observe that, when the control parameters belong to any of these regions, the stable manifold of ðg; 0Þ connects ð0; 0Þ and ðg; 0Þ along prey axis, while its unstable manifold separates the positive quadrant into two connected regions. This unstable manifold touches the y-axis and from there it continues to move on the y-axis heading towards þ1. Hence, those paths which lie below this unstable manifold eventually move towards the y-axis and reach it in a finite time under the influence of the unstable manifold and hence are likely to reach the y-axis at a lower level than that of the unstable manifold. This situation reverses for those paths initiating above the unstable manifold. Upon reaching the y-axis the solution moves on it in the increasing direction of y. Hence the system predicts unbounded growth for the predators when the control parameters belong to any of the regions A1 , A2 or A3 . Also note that, after the system becomes free from the prey, the predators can be maintained at any level by switching to the food supply x to d=ðb daÞ, after the predator population reaches the required level. Here, both the unlimited growth for the predators and the possibility to maintain the predators at any level with the same amount of food supply are somewhat unrealistic. This is due to our not assuming intra-specific competition among the predators at higher concentrations. From the above analysis, it can be observed that if the additional food supply level and its quality satisfies g 1 axo0
(21)
then the interior equilibrium of (11) is globally asymptotically stable. Otherwise, the stability of the interior equilibrium depends on the equilibrium prey population level. In fact, this analysis makes it possible for us to choose an appropriate additional food and its supply level so that the ecosystem can be driven to either a desired level or to a cycle about a specified equilibrium population. For, ~ yÞ ~ is the desired equilibrium state for the ecosystem, if ðx; then this will be admitted by the system (11) if a40 and x40 are chosen to satisfy ax þ 1 þ x~ ¼
~ b yg ¼ ðx~ þ xÞ. g x~ d
(22)
This equilibrium can be reached asymptotically if (21) is satisfied. On the other hand, if g 1 ax40
(23)
~ y~ can be reached asymptotically if then x; ~ x4
g 1 ax 2
(24)
else the state will reach a limit cycle asymptotically which is ~ yÞ. ~ centered around ðx;
4. Consequences of providing additional food In this section, we study the effect of providing the predator with additional food on the ecosystem dynamics. As observed in the previous section this effect is twofold. The system takes two opposite directions depending on the position of a relative to b=d. We characterize the additional food to be of high quality if aob=d and it is of low quality if a4b=d. Originally, if the ecosystem supports coexistence of prey and predators (i.e., the system (10) admits an interior equilibrium point whether stable or unstable) then continuous supply of high-quality additional food to the predators with the supply level x 2 ½0; d=ðb daÞÞ decreases the equilibrium prey population from d=ðb dÞ. Increasing the additional food supply beyond this interval eradicates the prey from the ecosystem in a finite time, and from that time the predators survive only on the external food supply. From this, we conclude that high-quality additional food source favors the biological control by benefiting the predator enough to suppress the target prey. On the other hand, providing predators with low-quality additional food the equilibrium prey–predator population moves from ðx ¼ d=ðb dÞ; gðx ÞÞ and heads towards ðg; 0Þ as the food supply x increases in ½0; ðgðb dÞ dÞ= ðda bÞÞ. Increasing the food supply beyond the above interval leads the prey population to its carrying capacity and eliminates the predator from the ecosystem. Thus lowquality additional source does not benefit the biological control. On the contrary, it benefits the prey as it reduces predation pressure up on the prey. In fact, if the added food is very dense and the handling time for the food item is high, predator individuals encounter the food effectively to the exclusion of the prey. However, because handling time for the added food is so large, the predator cannot consume food fast enough to equal or exceed the predator death rate. The predator therefore goes extinct and the prey moves to its equilibrium density. The theory predicts such a situation as the predators are not assumed to be optimal foragers. In the case where a ¼ b=d, if in the absence of additional food the system has stable coexistence then increasing the food supply increases the predator equilibrium level but does not change the equilibrium level of prey population or the nature of the interior equilibrium. On the other hand, if the system is oscillatory in the absence of additional food then these oscillations can be eliminated with higher level of food supply, i.e., x4½ðb þ dÞ ðb dÞgd=bðd bÞ. From the control parameter space analysis we observe that the additional food characteristics i.e., predator’s handling time for the additional food plays a very important role in the controllability of the ecosystem. With suitable choice on the additional food and its supply level the ecosystem can be made either prey dominated or predator dominated. The system can also be steered to a desired level subject to the parameter values satisfying required conditions. If a specific choice of food, supplied at
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a chosen level, drives the system oscillatory then we can eliminate such fluctuations by choosing a substitute for the additional food which stabilizes the system by eliminating the oscillations. Thus, through additional food we can eliminate either the predator alone and allow the prey to reach its carrying capacity or eliminate prey alone and subsequently maintain the predators in the ecosystem with the additional food. We can eliminate both prey and predator altogether by withdrawing the additional food after the prey gets eradicated from the ecosystem. 5. Numerical illustration In this section we illustrate some of the key findings using numerical simulation. We assume the values of the ecosystem parameters g, b and d to be 6.0, 0.3 and 0.2, respectively. Clearly these values satisfy (17) indicating existence of a unique asymptotically stable limit cycle for the system (10) about the equilibrium ð2:0; 2:0Þ. A path with initial value ð1; 1Þ approaching this limit cycle is shown in Fig. 4. Here we have b=d ¼ 1:5. In the simulations to follow we wish to illustrate the controllability of the system (11). ~ yÞ ~ to be (3.0, 3.0). From (22), Now, let us fix the target ðx; we have the values of a and x to be 2.0 (4b=d) and 1.0 which satisfy (24). Therefore, it is possible to reach the target by choosing the quality of additional food and its quantity as 2.0 and 1.0, respectively. Fig. 5A presents the path of the system (11) which initiates at ð1; 1Þ. This illustrates the possibility to eliminate cycles in the ecosystem through provision of additional food to predators. Observe that we are moving from the region B32 into C 32 in Fig. 3 with the above choices for a and x. Fixing the target population level to be (1.0, 3.0), we obtain the values of a and x to be 1.1429 (ob=d) and 1.4 (from (22)). As these parameter values do not satisfy (24),
Fig. 4. Numerical illustration showing predator and prey evolution in the absence of additional food.
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the state of the system cannot reach the target but tends to a limit cycle about the target. This illustrates the change in the dynamics due to moving upwards in the region B31 from a-axis in Fig. 3. Path of the system (11) initiating at ð1; 1Þ is presented in Fig. 5B. Observe that providing additional food, with the specifications mentioned above, compresses the limit cycle and moves it towards the y-axis. Suppose we wish to retain the prey population at 2:0 only but increase the predator population to 5:0. From the analysis presented in the previous sections, we have a ¼ b=d ¼ 1:5 and from Eq. (22) we obtain x ¼ 3:0. Since these parameter values satisfy (24), we have the target state ð2:0; 5:0Þ to be an asymptotically stable equilibrium. Fig. 5C presents path of the system (11) which illustrates the approach to the target state in the presence of additional food. If we fix the target level to be ð2:0; 2:45Þ, as earlier we have a ¼ 1:5 and from (22) we obtain x to be 0:45. With these parameter values we approach a limit cycle about ð2:0; 2:45Þ asymptotically as (24) is not satisfied by the control parameter values. Here, the parameter values lie on the line a ¼ b=d ¼ 1:5 with x ¼ 0:45 2 ð0; 23Þ ¼ ð0; ½ðb þ dÞ=ðb dÞgd=bðd bÞÞ. The corresponding simulation for the system (11) is shown in Fig. 5D. 6. Discussion and conclusions In spite of growing prominence in practical biological control as well as theoretical study for studying the consequences on predator–prey dynamics due to provision of additional foods to the predators, there is increasing need to bridge these two areas explicitly. To improve our understanding on these aspects, it is important to have a closer look at various mechanisms, assumptions and hypotheses involved in the theoretical models, built to represent predator–prey dynamics in the presence of additional food to predators. In this paper, we make a systematic analysis of the dynamics of a predator–prey system when some additional food is provided to the predator. The additional food is assumed to be either non-reproducing prey or some food source. We do not make any distinction regarding the additional food like complementary, substitutable, essential or alternative (Hodek and Honek, 1996; Sabelis and van Rijn, 2005). We only assume that the predator is capable of reproducing by consuming either of the available food, and we discuss about the food quality only when relevant to understanding the model or predictions. We characterize quality of a food with respect to its handling time. The dynamics of the predator–prey system in the presence of additional food is viewed as a variation of predator–prey model with Holling type II functional response. From the analysis of the model, we observe that the ratio of predator handling times for the additional food and prey plays a key role in the controllability of the ecosystem. We have three cases here. In the first case we have the ratio of handling times mentioned above is greater than the ratio between the
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Fig. 5. Numerical illustrations showing predator and prey evolution in the presence of additional food with different handling times and supply levels.
maximum predator growth rate and its starvation rate. We have the above relation reversed in case 2. In case 3, we have the ratios to be equal. In the first case, we clearly have the handling time for the additional food to be higher than that of prey. Here, we find that, if the predators go extinct in the absence of additional food, it is not possible to bring in eventual predator–prey coexistence by providing additional food to predators. Therefore, if in the absence of additional food, the births due to consumption of prey are not compensating the deaths, then adding any amount of additional food cannot improve the situation. This is due to the distraction effect caused by the addition of low-quality food which takes more time to consume when compared with that of a prey item and the predators are time limited. Thus, in this situation biological control is not possible through additional food. Instead, if the predators and prey have a stable coexistence in the absence of additional food, then the
distraction caused by the provision of additional food to the predators partially relieves the prey from predation pressure, and also decreases the birth rate of the predator due to higher value of the handling time for the additional food. As the predators encounter the additional food they spend more time in consuming the additional food and a balancing quantity of the prey is released. This observation goes with the inferences made by Harwood and Obrycki (2005), wherein it was concluded that—Although conservation biological control may enhance predator growth rate by providing an abundant and nutritionally balanced diet, it is feasible that predators will divert feeding efforts towards non-pest food items, thus reducing biological control. Also, increase in the additional food (greater than a critical value) decreases the birth rate of the predator. Hence increase in the concentration of such additional food increases the distraction effect causing a decline in the predator population and increasing the prey population. Thus the coexistence continues as long as the additional
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food level is less than a critical value beyond which the net births due to consuming the available food does no longer compensate the deaths resulting in elimination of predators, and the prey reach their carrying capacity. In this case, if the predator–prey system is eventually oscillatory in the absence of additional food, then the increase in the prey population due to additional food reduces the amplitude of the oscillations and beyond a concentration level the ecosystem will enter stable coexistence. The effects due to further increase of additional food level from here will be as discussed above. For the second case, where the ratio between the handling times of the additional food and the prey is less than the ratio between the maximum predator growth rate and its starvation rate, several subcases are possible. If the predator goes extinct in the system without additional food, then we can bring in coexistence of predator and prey by providing the predator with additional food with concentration level belonging to a specified interval. Here, the additional food increases the consumption of the predator population which reflects in the numerical abundance of the predator which in turn increases the per capita consumption of prey. As a result, there is a decrease in the equilibrium prey population. If, in the absence of additional food, the system has stable coexistence then providing additional food may either cause monotonic increase in the eventual value of the predator while maintaining stability, or it may bring in oscillations into the system when the supply goes beyond a specified level, leading to a stable limit cycle. These cycles move towards the predator axis with increase in the additional food supply. Beyond a certain level of food supply, the prey population goes extinct and predators are solely supported by the additional food. Even in the case where the eventual predator population is monotone with respect to food supply, the prey gets eradicated beyond a specific level of food supply. At this stage the predators can also be eliminated by with-drawing the additional food supply to the predators. If originally the predator and prey coexist in the absence of additional food but are oscillatory, then supplying the predators with additional food with sufficiently low handling time draws these oscillations closer to the predator axis in such way that beyond some supply level the prey population crashes and only predators survive eventually. Also it is possible to compress the oscillations and even eliminate them leading to their coexistence by providing food with relatively higher handling time. Whatever be the initial alterations in the behavior of the population due to increase in the additional food, increasing the food supply beyond a value always eradicates the prey from the ecosystem. It is important to note that even if the prey is of poor quality when compared to the additional food, the prey can be controlled biologically. This supports the inferences in Harwood and Obrycki (2005)—Simply because pests are a poor quality prey item (Toft, 2005) does not necessarily
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translate to little or no biological control in the field where generalist predators are frequently in a state of hunger e.g., (Bilde and Toft, 1998) and readily consume prey (Harwood et al., 2004, 2005). On the contrary, if the additional food happens to be of poor quality than prey, even then the prey population can be biologically controlled as long as it satisfies this second case. In the third case, if in the absence of additional food, the system admits an interior equilibrium, then introduction of additional food leaves the prey equilibrium level unaltered and the predator equilibrium increases with food supply. The nature of the equilibrium depends on the nature of the interior equilibrium in the absence of additional food. If it is asymptotically stable when there is no food supply, then it will continue to be asymptotically stable with increase in food supply. On the other hand, if it is unstable when there is no food supply, then the amplitude of the oscillations due to instability of the interior equilibrium decreases with increase in the food level and after a critical food level, the equilibrium turns asymptotically stable and it will remain in that state for all higher food levels. Finally, we conclude that the modified version of predator–prey model with Holling type II functional response, which represents predator–prey dynamics in the presence of additional food exhibits very interesting dynamics. Theoretically it is possible to control the dynamics of the system by manipulating the quality of the additional food and its supply level. It is possible to drive the prey and predator population to a desired level within specified limits. Also, it is possible to control and eliminate any oscillations present in the ecosystem by an appropriate choice of additional food and supply level. Oscillations can also be brought into the ecosystem through provision of additional food, if they are not present prior to the supply of food. We can not only eliminate prey by enhancing the predation effect by supplying the right kind of additional food to the predator, but also eliminate the predator by distracting the predator with a supply of low-quality additional food at high density which decreases the per capita growth rate of the predator below its starvation rate and releases the prey from predation pressure. In the context of biological pest control, the results caution the manager on the choice of quality and quantity of the additional food used for this purpose. An arbitrary choice may have completely opposite effects leading to increase in pest concentration and eradication of the predator. The analysis suggests a necessity to incorporate intra-specific competition for the predators at higher population level to make it more realistic.
Acknowledgments The authors are thankful to the three reviewers for their extremely valuable suggestions.
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