Dynamics of additional food provided predator–prey system with mutually interfering predators

Accepted Manuscript Dynamics of additional food provided predator-prey system with mutually in‐ terfering predators B.S.R.V. Prasad, Malay Banerjee, P.D.N. Srinivasu PII: DOI: Reference:

S0025-5564(13)00215-0 http://dx.doi.org/10.1016/j.mbs.2013.08.013 MBS 7405

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Mathematical Biosciences

Please cite this article as: B.S.R.V. Prasad, M. Banerjee, P.D.N. Srinivasu, Dynamics of additional food provided predator-prey system with mutually interfering predators, Mathematical Biosciences (2013), doi: http://dx.doi.org/ 10.1016/j.mbs.2013.08.013

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Dynamics of additional food provided predator-prey system with mutually interfering predators B.S.R.V. Prasada , Malay Banerjeeb , P.D.N. Srinivasu∗,c a Fluid

Dynamics Division, School of Advanced Sciences, VIT University, Vellore - 632014, India b Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur 208016, India c Department of Mathematics, College of Science and Technology, Andhra University, Visakhapatnam - 530003, India

Abstract Use of additional/alternative food source to predators is one of the widely recognised practices in the field of biological control. Both theoretical and experimental works point out that quality and quantity of additional food play a vital role in the controllability of the pest. Theoretical studies carried out previously in this direction indicate that incorporating mutual interference between predators can stabilise the system. Experimental evidence also point out that mutual interference between predators can affect the outcome of the biological control programs. In this article dynamics of additional food provided predator-prey system in the presence of mutual interference between predators has been studied. The mutual interference between predators is modelled using BeddingtonDeAngelis type functional response. The system analysis highlights the role of mutual interference on the success of biological control programs when predators are provided with additional food. The model results indicate the possibility of stable coexistence of predators with low prey population levels. This is in contrast to classical predator-prey models wherein this stable co-existence at low prey population levels is not possible. This study classifies the characteristics of biological control agents and additional food (of suitable quality and quantity), ∗ Corresponding

author; Tel: +91-891-2844703; Fax: +91-891-2755324 Email addresses: [email protected] (B.S.R.V. Prasad), [email protected] (Malay Banerjee), [email protected] (P.D.N. Srinivasu)

Preprint submitted to Mathematical Biosciences

September 3, 2013

permitting the eco-managers to enhance the success rate of biological control programs. Key words: Additional food, Biological control, Predator-Prey, Mutual interference, Functional response, Beddington-DeAngelis

1. Introduction Most important initiatives of the 20th century in the field of applied ecology has been, the control of populations of economically damaging species, particularly of agricultural weed and insect pests [28, 51]. Use of insecticides to control the pest has proved to have long-lasting side effects on the ecosystems viz., spraying pesticides contaminate the environment, further they get absorbed and remain as residue in the biomass of coexisting species in the ecosystem. Therefore bioremediation and biological control programmes have been envisaged to be an environmental friendly alternative for the use of insecticides which are found to damage the fragile ecosystems and deteriorate the environment.

In this context, lot of experimental work is being carried out in the area of biological control from various perspectives viz., finding optimal strategies to release natural enemies in the environment to control the pest, studying the effect of non-prey food items in enhancing the longevity, fecundity and ability of predators to control the pest [65, 66, 80, 101, 105]. For example, Macrolophus pygmaeus is found to be an efficient natural enemy for the control of white flies and other small arthropod pests in Europe [69, 75, 76, 77, 97]. Eggs of E. Kuehniella are used for rearing, mass production of these predators and subsequent release to control the whiteflies and other pests [29, 97]. Since, rearing these predators on eggs alone is an expensive option, experiments have been conducted [97] to determine whether pollen can be used as an additional food for nurturing the predators. It is found that the eggs and pollen is a better diet combination for optimal development of these predators [29, 97] instead of the diet being the eggs alone. Results of several other experiments conducted for 2

assessing the benefits of using additional food supplements in biological control programs can be found in [10, 26, 61, 67, 79, 92, 95, 103, 104, 105].

Some of the findings from the experimental works are supported by theoretical studies [19, 23, 24, 25, 82, 96] and more theoretical investigations are needed to explain the observations made in the field studies, in particular use of biotic/abiotic food items as an additional resource to predators and their efficacy in achieving the biological control targets [105]. In the recent contributions [88, 89, 90, 91], an attempt has been made to provide solutions for some practical problems, particularly, deriving control strategies in integrated pest management. The present study is focussed to provide solutions for a few more problems, such as combined effects of intra-specific competition between predators in presence of some additional food source.

It is well-known that the non-prey food sources, particularly plant materials, are integral part of the life history of predators and parasitoids [68]. Ecological significance of feeding on these non-prey food sources has long been recognised and attempts have been made to manipulate these non-prey sources (viz., nectar, pollen etc.,) in agricultural lands to enhance levels of biological pest control by natural enemies [20, 22, 38, 98]. W¨ ackers and van Rijn [102] point out that additional food (food supplements) in plant-herbivore-carnivore interactions is not only an important topic in basic ecology, but is also directly linked to the applied discipline of biological pest control.

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/additional food. This is modelled 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 which

3

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 [50, 96, 99]. This phenomenon, i.e., the presence of alternative food for a predator leading to reduction in equilibrium prey densities is termed as apparent competition [48].

A lot of theoretical work has been done in assessing the effects that can occur among the prey that share a predator [41, 49, and references there in]. These theoretical models concluded that introducing and manipulating with the non-pest alternative prey leads to apparent competition, due to which prey population is controlled eventually. However, empirical studies [44, 50, 70, 106] indicate that alternative prey/food provision need not always lead to successful biological control, bringing an apparent conflict between theory and empirical observations. It is remarked in [84] that success of biological pest control depends on the role of additional food, whether it is substitutable or complementary to the prey. Harwood and Obrycki [44] point out that provision of non-pest species to a (generalist) predator is twofold; on one hand these nutritious food items improve the predator population by enhancing their fecundity and on the other hand presence of these alternative food, result in reduced levels of pest consumption per individual predator.

To resolve the conflict between the theoretical studies and empirical observations, a theoretical model has been suggested in [88] to study the role of additional food as a biological control agent in predator-prey systems. The said model describes the interactions between the prey and predator in presence of additional food by assuming Holling type II functional response for predators towards the available food. The system analysis revealed several interesting phenomena which seemed contra-intuitive but agreed with experimental observations [42, 43, 44, 94, 106], thus validating the conceived model. It is observed that quality and quantity of additional food play a vital role in the controllabil-

4

ity of the system. These findings agree with the observations made in the recent review [105] dealing with the effects of artificial food sprays on conservational biological control wherein it is emphasised that the success of biological control much depends on quantity and quality of food sprays.

While the analysis of the model presented in [88] brought out several results confirming the observations made by experimental biologists, it predicted unbounded growth for predators when they are solely supported by the additional food (in the absence of prey). Since such unbounded growth in predator species is far from reality there is a necessity to improve the proposed model presented in [88]. Basically a limiting behaviour can be incorporated into the predators which should take care of this unbounded growth. Therefore incorporating either intra-specific competition or mutual interference within the predators seems to be a possible solution. Mutual interference within species is common in nature and it plays a key role in structuring the community dynamics [30, 31, 34, 60, 81, 93, 100]. It has been observed that mutual interference between the predators has profound impact on the success of biological control programmes [46, 81]. Thus studying the effect of mutual interference between the predators when these predators are provided with additional food is practically relevant, which is the focus of the current study.

In this article mutual interference has been assumed among the predators and investigations are made to study the consequences of such an assumption on the dynamics of additional food provided predator-prey system. One of the ways to incorporate the mutual interference between the predators is to modify the functional response of the predators towards the available food [2, 62, 78, 87]. The two well-known functional responses that model mutual interference between predators are Beddington-DeAngelis functional response [8, 27] and ratiodependent functional response [3, 5]. The works of [4, 7, 18, 62, 74, 78, 87] present a few other forms of functional responses through which mutual inter-

5

ference between the predators can be modelled.

The ratio-dependent functional response was introduced in [5] and well studied in literature [3, 9, 14, 37, 40, 52, 53, 54, 59, 63, 64, 73, 107]. It received a huge amount of favour as well as criticism [1, 2, 3, 6, 11, 16, 35, 45, 85]. The standard models involving ratio-dependent functional response are criticised mainly because of their singular behaviour at low densities. The BeddingtonDeAngelis form of functional response has some of the same qualitative features as the ratio-dependent functional response but avoids the singular behaviour at low population densities [16]. Beddington-DeAngelis model tends to the ratiodependent model if the interference parameter is large, provided that predator density is not negligible [7]. Thus, the ratio-dependent model can be considered as the limit of Beddington-DeAngelis model when the interference becomes very strong. Several models involving the Beddington-DeAngelis models have been developed and studied from different perspectives [7, 15, 32, 55, 62, 71, 72, 87].

In the current study, we use Beddington-DeAngelis type of functional response to model the mutual interference between the predators. As in [88], it is assumed here that the additional food is not dynamic but maintained at a constant level and the encounters between the predators and additional food is proportional to the density of additional food. Similarly the quality of the additional food is defined as the ratio of handling times between the additional food and prey [88]. The model analysis revealed several interesting phenomena including some significant bifurcations. These bifurcations are dependent on the strength of mutual interference between the predators. When this strength is high (> 1) it is found that the system always has a stable-coexistence. In this case it is not possible to eradicate the prey from the system by providing additional food to predators with any type of additional food. Reduced mutual interference (< 1) enriches the system dynamics with interesting bifurcations. In this situation, the system exhibits ratio-dependent like dynamics for a specific

6

choice of quality and quantity of additional food. As a result, the system has a stable coexistence at low prey population. This observation is in contrast to the classical predator-prey theory wherein we observe limit cycles at low population levels [32, 83, 88]. Our results highlight the vital role of the mutual interference in stabilising the population even at low population densities when the predator is provided with additional food.

The article is structured as follows: In the next section, the model representing the dynamics of predator-prey system (with mutual interference among predators) in presence of some additional food to predators is introduced. The conditions for the existence of various equilibria for the considered system are investigated in Section 3. Section 4 presents the local stability analysis of various equilibria that the model exhibits. Nature of interior equilibrium and its global stability are discussed in Section 5. Section 6 presents the global dynamics of the considered predator-prey system. Controllability of the considered predator-prey system, treating quality and quantity of additional food as control parameters, is presented in Section 7. Section 8 highlights the consequence of providing additional food to predators with mutual interference among them. Section 9 illustrates some of the key findings through numerical simulations. Finally, section 10 presents discussion and conclusions.

2. The Model Let us consider the following Beddington-DeAngelis model [8, 27] representing the predator-prey dynamics with mutual interference among predators 
cN P N dN − = rN 1 − , dT K a + N + ρP bN P dP = − mP, dT a + N + ρP

(1) (2)

here N ≡ N (T ), P ≡ P (T ) denote the biomass of prey and predator respectively with N (0), P (0) ≥ 0. The parameters r, K respectively represent the 7

intrinsic growth rate and carrying capacity of the prey. m is the mortality rate of predators in the absence of prey. Let hN , hP represent the handling time of the predator per prey item and interaction time between predators respectively then c =

1 hN

stands for maximum rate of predation. eN , eP are the

constants (with dimensions mass−1 time−1 ) that would depend on factors such as the predator’s movement rate while searching to detect prey or other predators [17]. The parameter a is given by

1 eN hN

, which is the normalisation coefficient

that relates the densities of the predator and prey (in the absence of additional food) to the environment in which they interact [27, 33]. If ε ∈ (0, 1) represents the efficiency with which the food consumed by the predator gets converted into predator biomass then the maximum growth rate of the predator given by εc is represented by the parameter b. ρ =

eP hP eN hN

measures the strength of mutual

interference among the predators. The above model (1)-(2) is well-studied in the literature from various perspectives [7, 15, 16, 32, 55, 56, 57, 62, 71, 72, 87].

Now let us assume that the predators are provided with additional food of biomass A, which is uniformly distributed in the habitat. Assuming that the number of encounters per predator with the additional food is proportional to the density of the additional food, the following model 
cN P N dN − = rN 1 − , dT K a + αηA + N + ρP b(N + ηA)P dP = − mP, dT a + αηA + N + ρP

(3) (4)

with non negative initial conditions represents the dynamics of additional food provided predator-prey system wherein the functional response is assumed to follow Beddington-DeAngelis formulation towards the available food.

If hA and eA respectively represent the handling time of the predator per unit quantity of the additional food and constant that signifies the the predator’s movement rate while searching to detect the additional food, then we have

8

α=

hA hN

and η =

eA eN .

Clearly, α (the ratio between the handling times towards

the additional food and the prey), signifies the quality of the additional food relative to the prey. Here, we term the additional food to be of low quality if the ratio of handling times between additional food and prey is greater than the ratio between the maximum predator growth rate and its death rate and it termed as high quality food if the above inequality reverses [88]. Thus the quality of the additional food is decided by the the value of α relative to the ratio

β δ.

From the definition of η we can infer that it represents effectual ability

of the predator to detect additional food relative to the prey. Thus, the term ηA represents quantity of additional food perceptible to the predator relative to prey. If A is taken to be zero then the system (3)-(4) gets reduced to the system (1)-(2). We, now intend to study the dynamics associated with the system (3)-(4).

Before proceeding further with the analysis of the proposed model (3)-(4), we non-dimensionalise the models (1)-(2) and (3)-(4) to reduce the number of parameters as well as the complexity associated with the analysis. The following transformations x=

cP N ,y = , t = rT a ra

(5)

reduce the systems (1)-(2) and (3)-(4) to the following non-dimensionalised versions respectively 
xy x − , x˙ = x 1 − γ 1 + x + y

y˙ =

βxy − δy, 1 + x + y

(6)


xy x − x˙ = x 1 − , γ 1 + αξ + x + y

y˙ =

β(x + ξ)y − δy, 1 + αξ + x + y

(7)

d , with where ·≡ dt

b m K ρ ηA β = ,δ = ,γ = , = ,ξ = . r r a c a 9

(8)

Table 1: Details about the variables and parameters present in the systems (1)-(2), (3)-(4), (6) and (7)

Variable/

Dimension

Definition

Non-dimensionalised

Parameter

representation

T

time

Time

t = rT

N

biomass

Prey (Pest) density

x=

P

biomass

Predator density

y=

−1

r

time

Prey intrinsic growth rate

K

biomass

Prey carrying capacity

γ=

−1

Maximum rate of predation

β=

−1

Maximum growth rate of predators

−1

Predator mortality rate

δ=

Strength of mutual interference among predators

=

c b m

time

time time

ρ a

biomass

Ratio between the handling times towards the additional food and the prey

η

Effectual ability of the predators to detect additional food relative to the prey biomass

K a b r m r ρ c

Normalisation coefficient relating the densities of prey and predator to the environment in which they interact

α

A

N a cP ra

Quantity of additional food provided to the predators

α

ξ=

ηA a

Details about the dimensions of the variables occurring in the systems (1)(2), (3)-(4) and their corresponding non-dimensional representations considered in the systems (6) and (7) are presented in the Table 1. Here it is pertinent to note that the parameter ξ represents the normalised quantity of additional food perceptible to predator relative to prey.

The following lemma establishes boundedness of solutions of the considered system (7). Lemma 1. All solutions of the system (7) initiating in the interior of the positive quadrant of the state space are bounded.

10

Proof. Let us define w = x + β1 y. For any η > 0 consider 
xy x (x + ξ)y δ η dw − + ηw = x 1 − + − y + ηx + y dt γ 1 + αξ + x + y 1 + αξ + x + y β β 
ξy δ η x − y+ y =x 1− +η + γ 1 + αξ + x + y β β 2 η−δ ξy γ(1 + η) + y+ ≤ 4 β 1 + αξ + y γ(1 + η)2 η−δ ξ ≤ + y+ . 4 β  Choosing sufficiently small η(< δ), we obtain dw + ηw ≤ M dt


γ(1 + η)2 ξ . = + 4 

Through an application of Gronwall’s inequality [13], we obtain

0 ≤ w(t) ≤

M (1 − e−ηt ) + w(0)e−ηt . η

Thus, as t → ∞ we have 0 < w(t) <

M η

implying that all solutions of the

system (7) are bounded.

3. Existence of Equilibria In this section we analyse the model (7) and draw conclusions regarding the conditions for existence of various equilibrium solutions. Observe that the nontrivial prey nullcline of (7) is a hyperbola which intersects the positive x−axis at (γ, 0), with x =

γ(−1) 

as its vertical asymptote. Further, it can be easily ob-

served that whenever 0 <  ≤ 1 the prey nullcline is a continuous smooth curve   and (γ, 0) in the positive quadrant of xy−space. joining the points 0, 1+αξ 1− For  > 1, the non-trivial prey nullcline is strictly decreasing and lies between x=

γ(−1) 

and x = γ, passing through (γ, 0) with x =

γ(−1) 

as an asymptote.

Note that the prey isocline of the system (7) is an increasing function of   , γ . Thus, provision of additional food to predators both ξ and α in γ(−1)  11

causes upward displacement to the non-trivial prey nullcline over the interval   γ(−1) , γ in comparison with the non-trivial prey nullcline of the system (6).  It can also be observed that the hump of the prey non-trivial nullcline shifts towards y−axis with increase in either α or ξ or both.

The non-trivial predator nullcline of the system (7) is a function of α and ξ which is a straight line whose slope ( β−δ δ ) matches with that of (6). This null    δ−(β−δα)ξ . Note , 0 and y−axis at 0, (β−δα)ξ−δ cline intersects x−axis at (β−δ) δ that this nullcline may move to the right or left from the predator isocline of (6) as ξ increases from zero, depending on the relative magnitude of α with respect to

β δ.

While the systems (6) and (7) always admit two common equilibria given by E0 = (0, 0) and E1 = (γ, 0), depending on the parameters of the system (7),   ˜ = 0, (β−δα)ξ−δ - an there is an emergence of two other equilibrium points, E δ axial equilibrium point and an interior equilibrium point E ∗ = (x∗ , y ∗ ), where, 1 x = 2β ∗

y∗ =

 [β(γ − ξ) − (β − δ)γ]   2 + [β(γ − ξ) − (β − δ)γ] + 4 [δ − (β − δα)ξ + βξ] βγ ,

(β − δ)x∗ − δ + (β − δα)ξ . δ

(9) (10)

Figure 1 presents four distinct nullcline configurations for the existence of interior equilibrium and other equilibria. Observe that the non-trivial prey nullcline is non monotone for the case  ∈ (0, 1) and it is monotonically decreasing when  > 1. Where as the predator non-trivial nullcline is always monotonically increasing.

From the monotonicity properties of the nullclines with respect to α and ξ, it can be inferred that providing additional food may either decrease or increase

12

A


B E


Predator axis

Predator axis

E∗

∗

~ E

E1

E0

E1

E0

Prey axis

Prey axis

C

D




E

∗

~ E

E1

E0

E∗

Predator axis

Predator axis




E1

E0

Prey axis

Prey axis

Figure 1: Figure presenting graphical illustration of nullclines and existence of equilibria for four distinct scenarios. Here, the blue and green curves represent the prey and predator non-trivial nullclines. Equilibria are represented by red dots. The frames A and B of the figure present two possible situations for the case where the interference parameter
∈ (0, 1) and the frames C and D correspond to the case
> 1. Note the coexistence of ˜ in the frames B and D the interior equilibrium E ∗ with the axial equilibrium E

the equilibrium level of the prey population depending on whether α is less than or greater than

β δ,

it always enhances the equilibrium level of the predator when

compared with that of (6). Also, if (6) does not admit an interior equilibrium then (7) shall never admit an interior equilibrium for α >

β δ.

But, for α <

β δ,

the

system (7) will admit an interior equilibrium even if (6) does not admit one, provided the non negative parameter ξ satisfies ξ >

δ−γ(β−δ) β−δα .

For this choice of ξ,

the system continues to admit a unique interior equilibrium for α ∈ (0, βδ ) as  is positive. If 0 <  ≤ 1 then the system will be relieved of this interior equilibrium Further, for a choice of α and ξ satisfying (β − δα)ξ − δ > 0,   ˜ = 0, (β−δα)ξ−δ . the system (7) admits an axial equilibrium point given by E δ if ξ ≥

δ (β−δα)−β .

˜ coexists with the interior equilibrium E ∗ as long The boundary equilibrium E as the parameters satisfy (β−δα)ξ−δ−βξ < 0 (cf. frames B and D of Figure 1).

The study presented in this article is focussed on the consequences of pro-

13

viding additional food to predators and use of such provision in the context of biological control. Here the additional food is characterised by two parameters α (quality of the additional food) and ξ (representative of quantity of the additional food). The analysis presented so far highlights the role of these two parameters in the existence/non-existence of various equilibrium solutions for the system (7). This component has significant role to play in biological control. Since the number of the equilibria change with variations in the parameters α and ξ, it is relevant to give a pictorial representation for the existence of equilibria in the (α, ξ) space. Hence, considering α and ξ as control parameters and treating β, δ, γ and  as system parameters, we shall now discuss about the regions of existence of various equilibrium solutions in the (α, ξ) space assuming that β > δ.

From the discussion presented above, the dynamics of the system (7) can be better viewed under the following four conditions involving the existence of interior equilibrium for the system (6) along with the conditions expressing the strength of mutual interference between predators δ β−δ δ γ> β−δ δ γ< β−δ δ γ> β−δ

γ<

and 0 <  ≤ 1,

(11)

and 0 <  ≤ 1,

(12)

and  > 1,

(13)

and  > 1,

(14)

and the curves

β(γ + ξ) − δ(1 + αξ + γ) = 0,

(15)

βξ − δ(1 + αξ) = 0,

(16)

βξ(1 − ) − δ(1 + αξ) = 0,

(17)

14

I

II

IV

III

~

~

Figure 2: Figure representing various regions defined by the curves (15), (16), (17) and

the line α = βδ when the system parameters, β, δ, γ and
satisfy the conditions (11) (14). Eq. (17),(16) and (15) define the boundaries between A, B and B, C(C1 ∪ C2 ) and ˜ 1 ∪ D2 ), respectively. Refer to the text and Table 2 for characterisation C(C1 ∪ C2 ), D(D of each of the regions.

associated with the existence of equilibria for (7). Note that the conditions (11) and (12) respectively imply the nonexistence and existence of interior equilibrium when the strength of mutual interference is low in the absence of additional food to predators. Similarly, the conditions (13) and (14) respectively imply the non existence and existence of interior equilibrium when the strength of mutual interference is high in the absence of additional food to predators. For a given set of system parameters, (11) to (17) present a complete picture regarding the dependence of the existence of various equilibria on the control parameters.

It can be easily observed that the predator nullcline for the system (7) passes

15

Table 2: Table presenting the equilibrium solutions admitted by the system (7) in various regions depicted in Figure 2

Region(s)

Equilibria

˜ D, D1 , D2 , D

E0 , E1

C, C1 , C2

E0 , E1 and E ∗

B

˜ E0 , E1 , E ∗ and E ˜ E0 , E1 and E

A

˜ when the parameters satisfy the through the equilibrium points E1 , E0 and E equations (15), (16), (17) respectively. Apart from these three curves the line α=

β δ

being an asymptote of (15) and (16) also plays a vital role in judging the

existence or emergence of interior equilibrium for the considered system. For the sake of simplicity let us denote the region βξ − δ(1 + αξ + ξ) > 0 by A and, the region between the curves (16) and (17), (15) and (16) be denoted by B and C respectively. Let the region defined by β(γ + ξ) − δ(1 + αξ + γ) > 0 for γ ≤

δ β−δ

be denoted by D and let the region defined by β(γ + ξ) − δ(1 + αξ + γ) < 0 for

δ β−δ

˜ The line α = < γ be denoted by D.

β δ

divides the region D into two

subregions given by D1 and D2 respectively. For the case

δ β−δ

< γ, the region C

gets further divided into two subregions C1 and C2 by the line α =

β δ.

Figure 2

presents all the above mentioned regions and the Table 2 presents the equilibria admitted by the system for the pair of control parameters (α, ξ) belonging to each of these regions.

4. Local Stability Analysis In this section, we study the asymptotic stability behaviour of the equilibrium points of the system (7) which requires evaluation of the community matrix J at each of its equilibrium points. We have

J(x,y)

⎡ ∂ ⎢F (x, y) + x ∂x F (x, y) =⎣ ∂ G(x, y) y ∂x

16

⎤ ∂ x ∂y F (x, y)

G(x, y) +

∂ y ∂y G(x, y)

⎥ ⎦,

(18)

where 
y x − , F (x, y) = 1 − γ 1 + αξ + x + y β(x + ξ) − δ. G(x, y) = 1 + αξ + x + y

(19) (20)

Evaluating J(x,y) at each of the boundary equilibrium points E0 = (0, 0),   ˜ = 0, (β−δα)ξ−δ we obtain E1 = (γ, 0) and E δ

JE0

JE1

and

⎡ ⎢1 =⎣ 0

⎡ ⎢−1 =⎣ 0

⎡ ⎢ JE˜ = ⎣

⎤ 0 βξ 1+αξ

−δ

1−  1−

(21)

⎤ γ − 1+γ+αξ ⎥ ⎦, β(γ+ξ) 1+γ+αξ − δ

y˜ 1+˜ y +αξ

β y˜ 1+˜ y +αξ

⎥ ⎦,

ξ 1+˜ y +αξ

(22)

⎤ 

0 β˜ y − 1+˜ y +αξ

⎥ ⎦.

(23)

From (21), (22) and the discussion presented in the previous section, we observe that the trivial equilibrium turns into an unstable node from a saddle and the axial equilibrium (γ, 0) turns into saddle from a stable node with the emergence of interior equilibrium. From (23) it can be inferred that the axial equilibrium (0, y˜) is a saddle as long as it co-exists with the interior equilibrium and turns into a stable node when the interior equilibrium ceases to exist. It is important to mention here that one eigenvalue of JE0 and JE1 becomes zero when the parameters satisfy the condition βξ − δ(1 + α) = 0 and β(γ + ξ) − δ(1 + αξ + γ) = 0 respectively indicating occurrence of transcritical bifurcation at the respective equilibrium points.

The characteristic equation associated with the Jacobian evaluated at the

17

interior equilibrium E ∗ = (x∗ , y ∗ ) is given by   λ2 − Tr J 

(x∗ ,y ∗ )

  λ + Det J 

(x∗ ,y ∗ )

= 0.

(24)

It can be easily verified that the system (7) satisfies the Kolmogorov condi tions [86] at (x∗ , y ∗ ) making Det J (x∗ ,y∗ ) positive. Therefore, the interior  equilibrium is locally asymptotically stable if Tr J (x∗ ,y∗ ) < 0 and unstable  if Tr J (x∗ ,y∗ ) > 0.

We have   Tr J 

(x∗ ,y ∗ )



β(x∗ + ξ)y ∗ 2x∗ (1 + y ∗ + αξ)y ∗ − = 1− − γ (1 + x + y ∗ + αξ)2 (1 + x + y ∗ + αξ)2 (25)

to be negative whenever the slope of the prey-nullcline is negative, since the first term in RHS of the above equation is nothing but the slope of the prey-nullcline of the system (7). Thus the interior equilibrium point will be asymptotically stable whenever  > 1 or when the predator isocline intersects the prey isocline at a point lying to the right of the hump of the prey isocline (which is likely to exist for the case 0 <  ≤ 1).

The expression in the RHS of (25) can be further simplified to obtain the following   Tr J 

(x∗ ,y ∗ )



 1 x∗ x∗ ξ δ − = 1− − − , γ β 1 + x∗ + y ∗ + αξ γ

(26)

from which it can be inferred that whenever the parameters of the system satisfies β ≥ 1, the associated interior equilibrium point is locally asymptotically stable.

The interior equilibrium point looses its stability through Hopf-bifurcation

18

  at the interior equilibrium when Tr J 

(x∗ ,y ∗ )

= 0 and this condition can be

obtained as the following quadratic equation in x∗ .   Tr J 

(x∗ ,y ∗ )

  βδξ ∗ 2 = H(x∗ ) ≡ − β(β − δ) + δ(1 − β)(β − δ) + x γ     − δ(1 − β) (β − δα)ξ − δ − βδξ    + β (β − δα)ξ − δ − δ(1 − β)(β − δ)γ − βδξ x∗     + δ(1 − β) (β − δα)ξ − δ − βδξ γ = 0.

(27)

Considering α as the bifurcation parameter one can verify the transversality condition for Hopf-bifurcation as shown in the appendix. It can be easily observed that the above quadratic equation (27) admits either two roots (H1 , H2 with H1 < H2 ) with identical sign or no roots. Therefore, the interior equilibrium E ∗ = (x∗ , y ∗ ) of the system (7) is stable whenever (27) does not admit any roots or x∗ ∈ (H1 , H2 ) and it is unstable if x∗ ∈ (H1 , H2 ). In view of the lemma 1, we clearly observe that the considered system admits a limit cycle whenever x∗ ∈ (H1 , H2 ).

5. Global Stability of the Interior Equilibrium In this section, we shall establish the asymptotic behaviour of the solutions of (7) in the positive quadrant of the phase space. We shall show that the interior equilibrium is globally asymptotically stable whenever it is locally asymptotically stable. For the sake of simplicity we denote A¯ = 1 + αξ, g = d1 =

δ β.

1 γ

and

Now, the system (7) can be rewritten as x (t) y , = (1 − gx) − ¯ x(t) A + y + x x ξ y  (t) = ¯ + − d1 . βy(t) A + y + x A¯ + y + x

19

(28) (29)

The interior equilibrium point (x∗ , y ∗ ) for the system (7) satisfies the following conditions: y = 0, 1 − gx∗ − ¯ A + y ∗ + x∗ x∗ ξ + ¯ − d1 = 0. A¯ + y ∗ +x∗ A + y ∗ +x∗

(30) (31)

Now, the following theorem establishes conditions for global stability of the interior equilibrium. Theorem 1. If (x∗ , y ∗ ) is locally asymptotically stable, then the system (7) has no nontrivial periodic orbit in R2+ . Proof. Let, if possible Γ(x(t), y(t)) be a nontrivial periodic orbit of the system (7) with period T > 0. We shall show that T tr J(Γ(x(t), y(t)))dt < 0,

(32)

0

which would contradict the stability property of the interior equilibrium.

In view of (28) and (29) we have T 

T tr J(Γ(x(t), y(t)))dt = 0

0

 x(t)y(t) −gx(t) + (1 − β) ¯ dt (A + y(t) + x(t))2 T − 0

βξy(t) dt. (A¯ + y(t) + x(t))2

(33)

Clearly (32) holds whenever 1 − β ≤ 0. Thus in what follows we assume 1 − β > 0. In view of the equations (30) and (31), a tedious but straight forward calculation yields

20

T tr J(Γ(x(t), y(t)))dt 0

 T T  x(t)y(t) βξy(t) −gx(t) + (1 − β) ¯ dt − dt = (A + y(t) + x(t))2 (A¯ + y(t) + x(t))2 0

0

T =

tr J(x∗ , y ∗ )dt +

0

T

(1 + d1 (1 − β))g(x∗ − x(t))dt

0

T − 0

ξ(g(A¯ + y ∗ ) + 1) (x∗ − x(t))dt ¯ (A + y ∗ + x∗ )(A¯ + y(t) + x(t))

T +

(A¯ +

0

T + 0

T − 0

y ∗

ξ(1 − gx∗ ) (y(t) − y ∗ )dt + x∗ )(A¯ + y(t) + x(t))

βξ(1 − gx(t)) dt − A¯ + y(t) + x(t)

T 0

(1 − β)g x(t)  y (t)dt β y(t)

x (t) (1 − β) ¯ dt − A + y(t) + x(t)

T 0

βξy(t) dt ¯ (A + y(t) + x(t))2

(34)

Using Eqs. (28) to (31), we obtain x (t) = x(t)


1 y∗ (x(t) − x∗ ) −g + ¯ A + y ∗ + x∗ A¯ + y(t) + x(t) A¯ + x∗ 1 + ¯ (y ∗ − y(t)), A + y ∗ + x∗ A¯ + y(t) + x(t)

(35)

and δ β−δ y  (t) (x(t) − x∗ ) + ¯ (y ∗ − y(t)). = ¯ y(t) A + y(t) + x(t) A + y(t) + x(t)

(36)

Therefore we have   x (t) y  (t) A¯ + y(t) + x(t) , δ − A2 x − x(t) = x(t) y(t) A3 + δg(A¯ + y(t) + x(t)) ∗



21

(37)

and   x (t) y  (t) A¯ + y(t) + x(t) y  (t) ¯ , d − A1 + g(A + y(t) + x(t)) y −y(t) = x(t) y(t) y(t) A3 + δg(A¯ + y(t) + x(t)) (38) ∗

where A1 =

y∗ ∗ +x∗ , A2 ¯ A+y

=

∗ ¯ A+x ∗ +x∗ , ¯ A+y

d = β − δ and A3 = A2 d − A1 δ.

Now, using (37), (38) and by repeated application of Green’s theorem [36, 58] we obtain T tr J(Γ(x(t), y(t)))dt 0

T

∗

∗

2



tr J(x , y )dt − (1 + d1 (1 − β))gδ A3

= 0

 − (1 + d1 (1 − β))gA2 A3 δξ(g(A¯ + y ∗ ) + 1) − A¯ + y ∗ + x∗ − − − −



Ω

Ω

Ω

1 dxdy [A3 + δg(A¯ + y + x)]2 x

1 dxdy ¯ [A3 + δg(A + y + x)]2 y

δ2 g dxdy [A3 + δg(A¯ + y + x)]2 x

 δg A2 ξ(g(A¯ + y ∗ ) + 1) dxdy A¯ + y ∗ + x∗ [A3 + δg(A¯ + y + x)]2 y Ω  δ2 g dξ(1 − gx∗ ) dxdy ∗ ∗ ¯ A + y + x [A3 + δg(A¯ + y + x)]2 x Ω  δg A1 ξ(1 − gx∗ ) dxdy ∗ ∗ ¯ A + y + x [A3 + δg(A¯ + y + x)]2 y Ω  A3 gξ(1 − gx∗ ) dxdy A¯ + y ∗ + x∗ [A3 + δg(A¯ + y + x)]2 y Ω

T 1 βξy(t) dxdy + dt − β ξ (A¯ + y + x)2 (A¯ + y(t) + x(t))2 0 Ω   (1 − β)g 1 1 − dxdy dxdy − (1 − β) β y (A¯ + y + x)2 2



Ω

T − 0

Ω

βξy(t) dt (A¯ + y(t) + x(t))2

< 0.

(39)

22

Hence, the divergency criterion [39] implies all the periodic solutions must be orbitally stable. This is impossible, since (x∗ , y ∗ ) is locally asymptotically stable. Thus, the system (7) has no periodic orbit in R2+ .

6. Global Dynamics In this section, we study the global dynamics of the considered system with respect to variations in the (α, ξ) control parameter space. First, note that the Hopf bifurcation curve (27) can be rewritten in the parametric form involving α,ξ as 
√

   1 1 (β(γ − ξ) − (β − δ)γ) + ∆ δ δ − − − +1 β 2βγ β  √ ⎫ ⎧ ∆ ⎬ ⎨ (β − δα)ξ − δ + (β − δ)γ (β(γ−ξ)−(β−δ)γ)+ 2βγ × ⎭ ⎩ δ  −ξ

(β(γ − ξ) − (β − δ)γ) + 1− 2βγ

√ 2 ∆

= 0,

(40)

where ∆ = (β(γ − ξ) − (β − δ)γ)2 + 4(δ − (β − δα)ξ + βξ)βγ.

From the discussion presented in the section 3, it is clear that under the natural conditions (11)-(14), the curves (15)-(17) together with the above Hopfbifurcation curve reveals the global dynamics of the system (7) by treating α and ξ as control parameters.

Observe that the curves (17) and (40), representing transcritical bifurcation at (0, y˜) and Hopf bifurcation at (x∗ , y ∗ ), are dependent on the parameter  and valid only for 0 <  ≤ 1. The curves (15) and (16), representing transcritical bifurcation at (0, 0) and (γ, 0) respectively, are independent of the parameter . Note that α = α=

β δ

β δ

is a common asymptote for the curves (15) and (16). The line

−  is an asymptote for the curve (17). These asymptotic thresholds play

a significant role in the global behaviour of the considered system.

23

First, we shall discuss the global dynamics associated with the system when  > 1. As mentioned in the section 5, the interior equilibrium in this case, if exists, is globally asymptotically stable. Further, it can be inferred from the qualitative properties of the non-trivial prey nullcline for the system (7) that the prey population can not be eradicated from the system by the provision of additional food to predators. The dynamics in this case can be explained through the Frames II and IV of Figure 2.

From Frame II, observe that for a fixed α, as ξ increases from zero we may enter either D1 or D2 depending on the value of α. In either case the paths approach (γ, 0) asymptotically. In this region (0, 0) is the co-existing equilibrium with saddle nature. As we enter the region C from D1 , the interior equilibrium comes into existence with transcritical bifurcation taking place at (γ, 0). Here, the nature of (0, 0) remains unchanged. Moving into B from C we have the emergence of axial equilibrium point (0, y˜) on the predator axis as a consequence of transcritical bifurcation at (0, 0), which turns (0, 0) into unstable node and (0, y˜) to a saddle. In this region the nature of other co-existing equilibria (γ, 0), (x∗ , y ∗ ) remains unchanged.

From Frame IV, observe that for a fixed α, as ξ increases from zero we may enter either C1 or C2 depending on the value of α. In both the regions the system admits a globally asymptotically stable interior equilibrium along with an axial equilibrium (γ, 0) and the trivial equilibrium (0, 0). These two boundary ˜ the interior equiequilibria are saddle in nature. As we enter from C2 into D librium ceases to exist due to a transcritical bifurcation at (γ, 0) making the later asymptotically stable. In this region the nature of the trivial equilibrium (0, 0) remains unchanged. As we enter from C1 into B, the axial equilibrium (0, y˜) comes into existence as a result of transcritical bifurcation at the trivial equilibrium point (0, 0). As a consequence, (0, 0) turns unstable node and (0, y˜)

24

exhibits saddle nature. In this region the interior equilibrium continues to be globally asymptotically stable.

To study the global dynamics associated with the case 0 <  ≤ 1, we first need to recall the following observations. From the linear analysis done about the interior equilibrium we observed that the unique interior equilibrium is globally asymptotically stable as long as 1 − β ≤ 0. Thus the case 0 <  ≤ 1 can be further divided into two sub cases given by 0 <  <

1 β

and

1 β

≤  ≤ 1. Since

there is going to be no occurrence of Hopf bifurcation for the latter case the division of (α, ξ) parameter space for this case is as given in the Frames I and III of Figure 2 and this division would involve an additional curve (40) signifying occurrence of Hopf bifurcation for the case 0 <  <

Concentrating on the case

1 β

1 β.

≤  ≤ 1 and comparing the parameter space

division with the respective divisions presented for  > 1 (cf. Figure 2), we observe presence of transcritical bifurcation curve at (0, y˜) which brings another significant region A into the (α, ξ) parameter space. While the dynamics of the system remain the same in the cases

1 β

≤  ≤ 1 and  > 1 for the regions

˜ C1 , C2 , B (cf. Figure 2); the system experiences transcritical biD1 , D2 , C, D, furcation at the axial equilibrium (0, y˜) as we enter into the region A from B. As a result, the interior equilibrium point vanishes stabilising the axial equilibrium point (0, y˜). This behaviour is common to both the cases γ <

δ β−δ

and

δ β−δ

< γ.

˜ coincides for the cases The dynamics of the system in the regions D1 , D2 , D 0<<

1 β

and  > 1. While, the system experiences transcritical bifurcation

at the axial equilibrium (0, y˜) annihilating the interior equilibrium; the presence of Hopf bifurcation curve in B ∪ C divides it into either six or five or four subregions depending on the shape of the Hopf bifurcation curve, which is  dependent. This phenomenon is illustrated through the Figure 3 which presents the Hopf bifurcation curve for three significant values of . Essentially, this

25

Bifurcation plot for

I

Bifurcation plot for

II

Bifurcation plot for

III

Figure 3: Figure presenting the Hopf bifurcation curve (red) as a function of
along with the other bifurcations curves (cf. Figure 2) for the system (7). Frames I and II present the δ . The changes in the Hopf bifurcation curve for two significant
’s for the case γ > β−δ ecosystem parameters are assumed to be β = 0.6, δ = 0.3, γ = 10. Frame III presents the δ Hopf bifurcation curve for
= 0.07 for the case γ < β−δ .

26

Hopf bifurcation curve divides B ∪ C region into two parts in such a way that the interior equilibrium corresponding to points of one part would be globally asymptotically stable while those due to the points in the other part would be unstable leading to existence of asymptotically stable limit cycle.

The three frames shown in the Figure 3 present interesting situations where the system undergoes Hopf bifurcation twice by fixing one of the parameters and varying the other. The parameter to be fixed depends on the value of  (or the shape of the Hopf bifurcation curve). This phenomena can be observed in the Frame I (Frame II) of the Figure 3 on the line ξ = 1 (α = 0.3). Similar behaviour can be observed on the lines α = 0.2 and ξ = 1 of Frame III. Note that for the fixed ξ cases referred above, with increase in α, the system initially stabilises at points close to predator axis. With further increase in α the system undergoes Hopf bifurcation twice and stabilises again at points with relatively large prey component (cf. Frames I-III of Figures 4 and 5).

The above mentioned interesting situation is illustrated in the Figure 6. Observe the persistent large amplitude oscillation in prey population density for a range of values of α, which is clear from the lower and upper bound of the stable limit cycle as presented in the bifurcation diagram. An opposite situation occurs for the case where α is fixed and ξ is allowed to vary. Here, with increase in ξ the system stabilises initially at points with larger prey component when compared to those obtained after the occurrence of second Hopf bifurcation with further increase in ξ (cf. Frames IV-VI of Figures 4 and 5). The proximity of the stable equilibrium to the predator axis can be inferred from the proximity of the corresponding α, ξ parameter values to the transcritical bifurcation curve of (0, y˜).

27

28

y→

y→

x→

x→

x→

(α,ξ)=(0.3,0.5)

(α,ξ)=(2.0,1.0)

(α,ξ)=(0.5,1.0)

I

V

III

x→

x→

x→

(α,ξ)=(0.3,1.1)

(α,ξ)=(0.3,0.1)

(α,ξ)=(1.0,1.0)

VI

IV

II

Figure 4: Figure presenting the interesting situation where the system undergoes Hopf bifurcation twice by fixing one of the control parameter and changing the δ other for the case γ > β−δ with β = 0.6, δ = 0.3 and γ = 10. Frames I-III (IV-VI) presents the transition of the system from stability to instability and then back to stability when
= 0.3 (
= 0.45) and by fixing the control parameter ξ = 1.0 (α = 0.3) and changing the parameter α (ξ).

y→

y→ y→ y→

29

y→

y→

x→

x→

x→

(α,ξ)=(0.2,0.85)

(α,ξ)=(0.4,1.0)

(α,ξ)=(0.26,1.0)

V

III

I

x→

x→

x→

(α,ξ)=(0.2,0.95)

(α,ξ)=(0.32,1.0)

VI

IV

II

Figure 5: Figure presenting the interesting situation where the system undergoes Hopf bifurcation twice by fixing one of the control parameter and changing the δ other for the case γ < β−δ with β = 0.6, δ = 0.45, γ = 2.5 and
= 0.07. Frames I-III (IV-VI) presents the transition of the system from stability to instability and then back to stability by fixing the control parameter ξ = 1.0 (α = 0.2) and changing the parameter α (ξ).

y→

y→ y→ y→

5 4 3 2 1 0 0.5

1

1.5

2

Figure 6: Bifurcation in equilibrium density of prey for a range of values of α. Other parameter values are γ = 10, ξ = 1,
= 0.3, β = 0.6 and δ = 0.3. Green curve represents the attracting steady-state level for prey species (in case of stable limit cycle, upper and lower bounds are shown here) and the red broken line represents the steady-state density of prey population when interior equilibrium point is unstable.

7. Controllability In this section we study some controllability aspects pertaining to the system (7) with respect to the control parameters α and ξ. The analysis presented in the previous sections makes it possible for us to choose an appropriate additional food and its supply level so that the system can be driven to either a desired equilibrium state or to a limit cycle about a specified equilibrium state. For, if (ˆ x, yˆ) is the desired equilibrium state for the system, then (ˆ x, yˆ) will be admitted as the interior equilibrium for the system (7) if α > 0 and ξ > 0 are chosen to satisfy 1 + αξ + x ˆ + yˆ =

β yˆγ = (ˆ x + ξ). γ−x ˆ δ

(41)

This equilibrium (ˆ x, yˆ) can be reached asymptotically if 0 < x ˆ < H1 or H2 < ˆ ∈ (H1 , H2 ) then x ˆ < γ where H1 and H2 are the roots of the equation (27). If x the state will reach a limit cycle asymptotically that surrounds (ˆ x, yˆ). In this case the size of the limit cycle depends on the proximity of x ˆ to either H1 or H2 .

If it is needed to eradicate the prey from the system and maintain the predators at a specific level y˜ then the parameters of the system need to satisfy the conditions 0<ξ<

30

δ y˜, β

(42)

and βξ − δ(1 + αξ) + δ˜ y = 0.

(43)

8. Consequences of providing additional food to predators influenced by mutual interference In this section, we discuss the consequences of providing additional food to predators influenced by mutual interference and its effect on the system dynamics. From the discussions presented in the previous sections, we conclude that the effect of provision of additional food to predators is twofold. The system takes two opposite directions depending on the relative position of α with respect to

β δ.

value if α <

β δ

Thus, we characterise additional food to be of high nutritional and it is of low nutritional value if α >

β δ

[88, 89, 90, 91]. The

degree of quality is related to the ability of the predators to control the prey by consuming such a food.

Originally, if the system supports coexistence of prey and predators (i.e., the system (6) admits an interior equilibrium point (¯ x, y¯) whether stable or unstable) then the equilibrium prey population can be driven to levels lower than x ¯ if the predators are provided with food of high quality. The extent to which the prey population can be reduced depends not only on the quality and quantity of the additional food but also on the strength of mutual interference () among predators.

In order to eliminate the prey from the system, apart from providing high quality additional food satisfying α <

β δ

and quantity ξ >

δ β−δα−β ,

it is nec-

essary that the strength of mutual interference be low (0 <  ≤ 1). For higher values of this parameter ( > 1), the prey can not be eliminated from the system although their numbers can be driven to the vicinity of quantities of high quality additional food.

31

γ(−1) 

by providing high

It should be noted that, for a given quality (α <

β δ

− ) the system requires

more quantity of additional food to eradicate the prey from the system if the mutual interference is higher but less than one. After the eradication of prey from the system, the predator numbers can be controlled by manipulating the additional food. Hence, we conclude that high quality additional food favours the biological control by benefiting the predator (in spite of the mutual interference between predators) enough to suppress the target prey. It should be noted that the ability and the ease with which the prey gets suppressed depends on the magnitude of the mutual interference. Lower strength of mutual interference favours biological control.

On the other hand, providing predators with low quality additional food β δ ),

the equilibrium of the system heads towards (γ, 0) as the food supply   . This phenomena is observed for all  > 0. Increasξ increases in 0, γ(β−δ)−δ δα−β

(α >

ing the food supply beyond the above interval leads the prey population to its carrying capacity and eliminates the predator from the system. In this case, the prey seems to be getting benefited through the provision of high quantities of low quality additional food, resulting in reduction of predation pressure upon the prey. Therefore, low quality additional food does not favour biological control.

Now let us consider the case where the system does not support predatorprey coexistence in the absence of additional food (i.e, the system (6) admits no interior equilibrium). Here, coexistence can not be established by providing additional food of low quality (cf. Figure 2). On the other hand, provision of additional food of high quality with supply level ξ less than

γ(β−δ)−δ δα−β

brings

coexistence into the system. From this stage reduction of prey population or eradication of prey population requires same procedures presented for the previous case.

32

Table 3: Outcome of the study as management recommendations in the context of biological control

Range of α 0<α<

β δ

β δ

<α

Here P1 = a b

Condition on ξ

Outcome

ξ < P1

Eventual prey dominated system with extinction of predator population

P1 < ξ < P2 < P3

Co-existence of prey and predator with prey dominancea,b,c

P1 < P2 < ξ < P3

Co-existence of prey and predator with predator dominancea,b,c

P2 < ξ

Prey free system in which predators are survived exclusively by additional food

ξ < P1

Co-existence of prey and predator with prey dominancea,c

P1 < ξ

Eventual prey dominated system with extinction of predator population

δ−(β−δ)γ β−δα ,

P2 =

In this case P1 < 0 for

δ β−δα

δ β−δ

and P3 =

δ β(1−)−δα

<γ

In this case P3 < 0 for  > 1

c

For  > 1 the co-existence is always stable. The co-existence can be stable or oscillatory in the case of 0 <  ≤ 1 and depends on the other ecosystem parameters (For more details refer Section 4)

33

From the control parameter space analysis we observe that predators handling time for the additional food and the strength of mutual interference between predators play a vital role in determining the system dynamics and its controllability. By a suitable choice on the additional food and its supply level one can drive the system to prey dominated state or predator dominated state.

For the case where the mutual interference between the predators is low, the analysis presents interesting situations where the system undergoes Hopf bifurcation twice by fixing one of the control parameters and varying the other. This phenomenon enables stabilisation of the state either at low prey population level or high population level by manipulating the quality and quantity of the additional food appropriately. Further, it is also possible to drive the system to a desired equilibrium state subject to parameters of the system satisfying certain conditions. In the case where the mutual interference is small the system can also be steered to a prey free state and maintain the predators at a desired level with continuous provision of appropriate quantity of high quality additional food. Table 3 summarises the outcome of the current study as management recommendations in the context of biological control.

9. Numerical Simulations In this section we demonstrate some of the key findings using numerical simulation. We assume the values of the ecosystem parameters γ, β, δ,  to be 10.0, 0.6, 0.3 and 0.4 respectively. With this choice of parameters, the system (6) admits an unstable equilibrium (2.50, 3.75). Consequently, there exists a stable limit cycle surrounding (2.50, 3.75) (cf. Frame I of Figure 7). In the following simulations we demonstrate the controllability of the system (7) by treating α and ξ as control parameters.

Suppose, we wish to keep the prey population at the same level as that of original system and increase the predator population and fix our target equilib34

35

y→

(2,2)

(0.5,3.5)

(2,2)

(2.5,3.75)

x→

x → (α,ξ)=(0.5843,1.3421)

III

I

(0,7)

(2,2)

(2,2)

(2.5,4)

x→

x→

(α,ξ)=(0.6,2.7143)

(α,ξ)=(1.4004,0.1667)

IV

II

Figure 7: Numerical illustration showing predator-prey evolution in the presence of additional food with different supply levels and handling times.

y→

(α,ξ)=(0,0)

y→ y→

rium (ˆ x, yˆ) to be (2.5, 4). Now using (41), we obtain the corresponding (α, ξ) to be (1.4004, 0.1666). For this choice of parameters, we have (ˆ x, yˆ) to be unstable ˆ < H2 < γ where H1 = 1.1345 and H2 = 2.5978. The limit cycle as 0 < H1 < x surrounding the equilibrium (ˆ x, yˆ) is shown in Frame II of Figure 7.

If we want to reduce the prey population by fixing our new equilibrium (ˆ x, yˆ) to be (0.5, 3.5) then from (41), we obtain α and ξ to be 0.5843 and 1.3421 respectively. For these parameter values, the equation (27) admits two positive ˆ < H1 < H2 . Thereroots H1 = 0.7714, H2 = 1.7617. Observe here that 0 < x fore, (ˆ x, yˆ) is asymptotically stable. Frame III of Figure 7 presents a path of the system (7) originating at (2, 2) and reaching the target (ˆ x, yˆ). This example illustrates the possibility of driving the system to a stable coexistence with low prey population levels by providing high quality additional food to predators.

Finally, we wish to eliminate the prey population from the system and maintain the predator population at a level yˆ = 7. Following the analysis presented at the end of the section 7, we obtain ξ = 2.7143 by fixing α = 0.6. With this combination of α and ξ the state of system (7) is driven to axial equilibrium ˜ = (0, 7). The corresponding trajectory of the system (7) initiating at (2, 2) E ˜ is presented in Frame IV of Figure 7. and reaching E

10. Discussion and Conclusions The effect of providing additional food to predators and its consequences on predator-prey dynamics is a topic of interest to theoretical as well as experimental scientists [12, 21, 41, 44, 70, 84, 88, 89, 90, 91, 96, 99]. These studies are being done from the biological control perspective with a view to derive strategies for controlling pest using natural enemies. It is well established that the quality and quantity of additional food play a vital role in controlling the pest (prey) in the agro-ecosystems [20, 22, 38, 98, 105, and references there in]. Assuming that the predators functional response to the available food is 36

of Holling type II, a systematic study has been made [88] to investigate the response of predators towards prey when additional food is provided to predators. Significant results have been obtained which confirmed observations made by experimental scientists [42, 43, 44, 94, 105, 106]. This study further indicated that the dynamics of predators should involve a limiting term to avoid the unbounded growth predicted for predators [88].

In this article an attempt is made to improve the earlier proposed predatorprey dynamic model by incorporating mutual interference among the predators. One of the ways to incorporate this mutual interference within predators is to replace the Holling type II functional response with Beddington-DeAngelis functional response [8, 27, 87]. In contrast to Holling type II functional response [47], Beddington-DeAngelis functional response has an extra term y in the denominator, which denotes the mutual interference between predators [16]. The parameter  measures the magnitude of mutual interference among the predators.

As in [88], no distinction (such as complementary, substitutable, essential or alternative) has been made regarding the additional food and its quality is characterised by its handling time. Accordingly, additional food is termed as low quality if the ratio of handling times between additional food and prey is greater than the ratio between the maximum predator growth rate and its starvation rate and it is of high quality if the above inequality reverses [88].

The dynamics and controllability of the proposed model are analysed for two distinct cases distinguished by the strength of mutual interference. In the first case this strength is assumed to be greater than unity (signifying stronger mutual interference) and it is assumed to be in unit interval for the second case. In the former case, it is possible to control and limit the prey population by the predator with provision of high quality additional food to predators. However,

37

the high mutual interference in predators deters the prey from going extinct. This is due to distraction effect caused on predators due to higher mutual interference and availability of large quantities of high quality additional food. It is note worthy that in the absence of additional food if the prey can not support the predators towards coexistence due to its poor nutritive value, it is possible to bring in stable coexistence by provision of high quality food to predators and thus control the prey in the system although it can not be driven to extinction.

In the later case, where the interference between the predators is weak (less than 1), along with bringing in coexistence and controlling the prey by providing additional food to predators it becomes possible to eradicate the prey from the system by appropriate choice on quality and quantity of additional food and strength of interference between predators. In this case, providing the predators with additional food may either cause an increase in the eventual predator population and decrease eventual prey population or it may even bring in oscillations into the system depending on the characteristics of the high quality additional food.

For a chosen high quality additional food, if the population cycles eventually then by increasing the quantity of additional food not only the amplitude of these cycles can be reduced but the system can be stabilised at low prey population. This observation is in contrast to the other classical predator-prey models with Holling type II functional response, where in the interior equilibrium exhibits instability nature at low prey population density, which induces limit cycles in the system dynamics [32, 83, 88]. The occurrence of limit cycles in these cases can be accounted for, by the fact that the predators are not able to reproduce enough at lower prey concentrations.

In the considered model the additional food of high quality acts as supplement to the predators, enhancing their growth at low prey concentrations. But,

38

because of the self-limiting behaviour of the predators, this growth is limited. Thus, the provision of additional food (of high quality) stabilises the system at low prey populations. With further increase in the quantity of the high quality additional food the predators eradicate the prey from the system and get stabilised on the predator axis. In this case the eradication becomes possible due to the ability for the predators to spend more time with the high quality additional food which results in numerical abundance of predators and loosing less time due to mutual interference.

Now, we bring forward the essential similarities and difference that are observed in the prey predator dynamics with respect to Beddington-DeAngelis functional response and Holling type II functional response. In both the cases the effort needed to eradicate the predator from system remains the same. The solution is to provide the predators with low quality additional food. The quantity required to eliminate the predators from the system remains the same both the models.

Coming to the controllability of prey population using high quality additional food, it is observed that in case of Holling type II response, for a given high quality provision of additional food beyond a critical quantity eradicates prey from the system in a finite time and continuing the provision of additional food triggers unbounded growth in predators. In case of Beddington-DeAngelis response, for the foresaid quality and quantity of additional food the coexistence still continues and it requires provision of higher quantities of high quality additional food to eradicate the prey asymptotically and stabilise the predator on the predator axis. Therefore the proposed model successfully eliminates the unbounded growth observed for the model presented in [88].

Finally it can be concluded that while high mutual interference favours stable coexistence of the predator and prey, in case of low mutual interference the

39

additional food can be used as a control to either eliminate any of the species or to have coexistence.

Appendix: Transversality condition for Hopf bifurcation TrJ(x∗ ,y∗ ) can be written as quadratic function of x∗ as follows, 2

TrJ(x∗ ,y∗ ) = Ax∗ + Bx∗ + C

(44)

where 

βδξ A = − β(β − δ) + δ(1 − β)(β − δ) + γ     B = − δ(1 − β) (β − δα)ξ − δ − βδξ

 < 0,

   + β (β − δα)ξ − δ − δ(1 − β)(β − δ)γ − βδξ ,     C = δ(1 − β) (β − δα)ξ − δ − βδξ γ < 0.

(45)

(46) (47)

Now, δξγ ∂x∗ > 0, = ∂α 2 [β(γ − ξ) − (β − δ)γ] + 4 [δ − (β − δα)ξ + βξ] βγ ∂A = 0, ∂α ∂B = δ 2 (1 − β)ξ + βδξ > 0, ∂α ! ∂C = −γ δ 2 (1 − β)ξ < 0. ∂α

(48)

(49) (50) (51)

Thus, # ∂x∗ ∂B ∗ ∂C ∂ " TrJ(x∗ ,y∗ ) = (2Ax∗ + B) + x + . ∂α ∂α ∂α ∂α

40

(52)

Now, as x∗ > 0 and from the equations (48) - (51) it follows that # ∂ " TrJ(x∗ ,y∗ ) = 0, ∂α

(53)

and hence the desired transversality condition is satisfied.

Acknowledgments The authors are extremely thankful to the two anonymous honourable reviewers for their critical comments and invaluable suggestions. The corresponding author acknowledges the support received from NBHM, Government of India.

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Highlights • Predators functional response is assumed to be of Beddington-DeAngelis Type • Dynamics investigated treating mutual interference as a parameter • High mutual interference between predators save prey from extinction • Low mutual interference between predators favours biological control • investigated controllability by treating additional food properties as parameters