A focus on long-run sustainability of a harvested prey predator system in the presence of alternative prey

T.K. Kar; S.K. Chattopadhyay

doi:10.1016/j.crvi.2010.09.001

Résumé

Within the framework of a general equilibrium model we study the long-run dynamics of a prey-predator model in the presence of an alternative prey. Our results show that sustainability, i.e. a positive value of the population in the long run, essentially depends on individual harvesting efforts and digesting factors relative to alternative prey. A detailed bifurcation analysis evidences the richness of possible long-run dynamics. Our model clearly shows that the role of an alternative prey must be taken into consideration when studying prey-predator dynamics.

1 Introduction

The modelling of commercial exploitations of renewable biological resources is a challenging task, as it includes the non-linear interaction of biological, economic and social components. In recent past years, many researchers have paid attention to the management of renewable resources, considering different parameters and externalities. In the management of common property renewable resources (such as fisheries, forestry and wildlife) harvested by competing individuals, societies or countries, the problem known as “The tragedy of commons” after Hardin [1]. Later on Clark [2], Masterton-Gibbons [3], Conrad [4] discussed many issues and techniques on this subject. Clark [2] discussed the problem of non-selective harvesting of two ecologically independent populations obeying logistic growth. Kar and Chaudhuri [5] discussed non-selective harvesting of a multispecies fishery. There are also some papers on competitive fish-species and on prey-predator models (see Gordon [6]; Datta and Mirman [7]; Bischi and Kopel [8]; Dubey et al. [9]; Bischi et al. [10]). Most of the prey-predator models consider a single prey called the focal prey; however, many authors have emphasized that the presence of alternative foods can effect biological control through a variety of mechanisms. For example, the presence of one prey species can have negative effects on the population of another prey species, by allowing the population of a shared predator to increase, thus leading to higher predation rates upon both prey items. In contrast, alternative prey can also lower predation on focal prey because of predator preference for alternative prey resources (Abram and Matsuda [11]). In such instances, the alternative prey can have a positive effect on population densities of the focal prey. Many massive piscivores, including spiny dogfish in the North-West Atlantic, and Pacific hake, undertake extensive seasonal migrations on a spatial scale much larger than the habitat occupied by some of their prey, and the development of realistic models for these prey-predator systems will require consideration of alternative prey. Vence [12], Spencer and Collie [13] have considered alternative prey in their models.

Methods used for managing predators, and associated public perception, are a crucial consideration in developing effective systems of management for predators and prey. Controlling predator populations to reduce predation on a threatened or endangered species may be difficult to achieve. Keeping wolf and cougar (wild American cat) populations in check might require reducing alternate prey. Control of mule deer populations in Sierra mountains of California might be required in addition to reductions in cougar numbers to prevent extirpation (complete destruction) of Sierra Nevada big horned sheep populations. The Alberta Government has issued additional hunting permits for moose and deer in an attempt to reduce alternate prey for wolves, hoping to reduce wolf numbers and thereby predation pressure on caribou.

In this study, we develop a simple, two species predator-prey model that incorporates the effect of alternative prey. We consider logistic production function of prey and a Holling type-II predator functional response (Holling [14]) so that there may present multiple equilibrium population levels. In our model, the growth rate of prey is formulated as

\frac{d N}{d T} = r N (1 - N / K) - \frac{α_{1} N P}{a + N}

(1)

where N = N(T) = size of prey at time T, P = P(T) = size of predator at time T, K = environmental carrying capacity of prey, r = intrinsic growth rate of prey, α₁ = predation co-efficient, a = half-saturation constant.

The growth rate of predator population is taken as

\frac{d P}{d T} = \frac{β_{1} α_{1} N P}{a + N} + d_{1} P (1 - N / K) - γ_{1} P

(2)

where β₁ is a conversion factor (we assume β₁ < 1, since the whole biomass of the prey is not transferred to the biomass of the predator), d₁ is the digesting factor relative to alternative prey and γ₁ is mortality rate of predator population.

Here, as the focal prey population (N) increases, the predator uses less alternative prey and when N → K the mass of alternative prey consumed tends to zero, and conversely, as the focal prey decreases, the predators increase their feeding on alternative prey. If N → 0 then dP/dT → (d₁ − γ₁)P. Thus even in case of the extinction of the focal prey the predator population maintains its growth rate, varying linearly with its density.

Now we introduce dimensionless variables by the following substitutionN = ax, P = ray/α₁, T = t/rthen our Eqs. (1) and (2) become

\frac{d x}{d t} = x (1 - x / \frac{K}{a}) - \frac{x y}{1 + x}

(3)

\frac{d y}{d t} = \frac{β_{1} α_{1}}{r} \frac{x y}{1 + x} + \frac{d_{1}}{r} y (1 - x / \frac{K}{a}) - \frac{γ_{1}}{r} y

(4)

Letting $\frac{K}{a} = η, \frac{β_{1} α_{1}}{r} = β, \frac{d_{1}}{r} = d, \frac{γ_{1}}{r} = γ$ we can write the governing equations as

\frac{d x}{d t} = x (1 - x / η) - \frac{x y}{1 + x}

(5)

\frac{d y}{d t} = \frac{β x y}{1 + x} + d y (1 - x / η) - γ y

(6)

For considering the exploited prey-predator system, we introduce scaled harvesting efforts E₁ and E₂ for prey and predator, respectively and then the equations governing our model become

\frac{d x}{d t} = x (1 - x / η) - \frac{x y}{1 + x} - E_{1} x

(7)

\frac{d y}{d t} = \frac{β x y}{1 + x} + d y (1 - x / η) - γ y - E_{2} y

(8)

The Note is organized as follows. The existence of different equilibrium points are considered in section 2. Stability analysis and bifurcations are given in section 3. In section 4, detailed numerical simulations are given. The Note ends with a conclusion in section 5.

2 Equilibrium of the model

Letting $ϕ_{1} (x) = \frac{x}{x + 1}, ϕ_{2} (x) = (1 - x / η - E_{1}) (x + 1)$ andϕ₃(x) = βϕ₁(x) + d(1 − x/η) − (E₂ + γ)the governing equations (7) and (8) become

\frac{d x}{d t} = ϕ_{1} (x) [ϕ_{2} (x) - y]

(9)

\frac{d y}{d t} = y ϕ_{3} (x)

(10)

The prey zero growth lines are obtained by setting dx/dt = 0, which gives ϕ₁(x) = 0, y = ϕ₂(x) that is x = 0 (y-axis) and the curve y = ϕ₂(x). The curve y = ϕ₂(x) passes through the points (0, 1 – E₁) and (η(1 − E₁), 0).

We see that

{ϕ^{'}}_{2} (x) = (1 - x / η - E_{1}) + (1 + x) (- 1 / η) = 1 - E_{1} - 1 / η - 2 x / η .

Thus, ${ϕ^{'}}_{2} (0) = 1 - E_{1} - 1 / η$ , andϕ₂′(η(1 − E₁)) = − (1 − E₁ + 1/η)

Thus if η(1 − E₁) > 1, then ${ϕ^{'}}_{2} (0)$ and ${ϕ^{'}}_{2} (η (1 - E_{1}))$ are of opposite signs and so ϕ₂(x) has a local maximum between x = 0 and x = η(1 − E₁). The predator zero growth lines are obtained by setting dy/dt = 0 i.e. y = 0 (x-axis) and ϕ₃(x) = 0. Thus isoclines of predator other than x-axis are the zeros of ϕ₃(x).

These are given by

β ϕ_{1} (x) + d (1 - x / η) - (E_{2} + γ) = 0,

or,

- d x^{2} / η + (β + d - d / η - E_{2} - γ) x + d - E_{2} - γ = 0

(11)

The zeros of ϕ₃(x) are given by

x_{2}, x_{1} = \frac{η}{2 d} [(d + β - d / η - E_{2} - γ) \pm \sqrt{{(d + β - d / η - E_{2} - γ)}^{2} - \frac{4 d}{η} (E_{2} + γ - d)}]

(12)

Here we investigate the different cases for positive zeros of ϕ₃(x) to have interior equilibrium. From the above discussion we already have two boundary equilibria P₀(0, 0), P₁((1 − E₁)η, 0) while P₁ exists if E₁ < 1.

Case 1.

If $(d + β - \frac{d}{η} - E_{2} - γ) > 0,$ and E₂ + γ − d < 0, that is, if $E_{2} < M i n {(β - \frac{d}{η}) + d - γ, d - γ}$ , then ϕ₃(x) has only one positive zero given by

x_{2} = \frac{η}{2 d} [(d + β - d / η - E_{2} - γ) + \sqrt{{(d + β - d / η - E_{2} - γ)}^{2} - \frac{4 d}{η} (E_{2} + γ - d)}]

(13)

and the corresponding value of y is

y_{2} = ϕ_{2} (x_{2}) = (1 + x_{2}) (1 - \frac{x_{2}}{η} - E_{1})

, which lies in

R_{2}^{+}

if E₁ < 1 − x₂/η.

Thus, in this case, the trivial equilibrium P₀(0, 0) always exists, the boundary equilibrium P₁((1 − E₁)η, 0) exists if E₁ < 1 and the only interior equilibrium P₂(x₂, ϕ₂(x₂)) exists if E₁ < 1 − x₂/η and $E_{2} < M i n {(β - \frac{d}{η}) + d - γ, d - γ}$ .

Here it can be observed that if P₂ exists then P₁ also exists.

Case 2.

Let $(d + β - \frac{d}{η} - E_{2} - γ) < 0$ , E₂ + γ − d < 0. In this case only one positive interior equilibrium may be obtained if (d − γ) > 0, (β − d/η) < 0 i.e. if d > max{γ, βη} and $(d + β - \frac{d}{η} - γ) < E_{2} < (d - γ)$ .

In this case x = x₂ is a positive zero of ϕ₃(x) and P₂(x₂, ϕ₂(x₂)) is a positive interior equilibrium while E₁ < 1 − x₂/η along with the above conditions. Therefore, in this case, the trivial equilibrium P₀(0, 0) always exists, the boundary equilibrium P₁((1 − E₁)η, 0) exists if E₁ < 1 and the interior equilibrium P₂(x₂, ϕ₂(x₂)) exists if E₁ < 1 − x₂/η, d > max{γ, βη} and $(d + β - \frac{d}{η} - γ) < E_{2} < (d - γ)$ . Here also we see that the conditions of existence of P₂ suffice the existence of P₁.

Case 3.

Let

(d + β - \frac{d}{η} - E_{2} - γ) > 0, E_{2} + γ - d > 0

and

{(d + β - \frac{d}{η} - E_{2} - γ)}^{2} > \frac{4 d}{η} (E_{2} + γ - d)

More explicitly,

d - γ < E_{2} < d - γ + β - \frac{d}{η}

and

{(d + β - \frac{d}{η} - E_{2} - γ)}^{2} > \frac{4 d}{η} (E_{2} + γ - d)

for γ < d < ηβ, or,

d < γ < (β - \frac{d}{η} + d)

Then there are two positive zeros of ϕ₃(x) which are x₁, x₂ given in (12) and

ϕ_{3} (x) = \{\begin{array}{c} > 0 \\ < 0 \end{array} \begin{array}{c} f o r x_{1} < x < x_{2} \\ f o r 0 < x < x_{1}, x > x_{2}, \end{array}

i.e.

{ϕ^{'}}_{3} (x_{1}) > 0

and

{ϕ^{'}}_{3} (x_{2}) > 0

. In this case all four equilibria exist. The boundary equilibrium P₁((1 − E₁)η, 0) exists if the condition:(H1): E₁ < 1 is satisfied. One interior equilibrium P₂(x₁, ϕ₂(x₁)) exists if the following conditions are satisfied:(H2):

{(d + β - \frac{d}{η} - E_{2} - γ)}^{2} > \frac{4 d}{η} (E_{2} + γ - d), E_{1} < 1 - x_{1} / η

with γ < d < ηβ, or

d < γ < (β - \frac{d}{η} + d),

and the other interior equilibrium P₃(x₂, ϕ₂(x₂)) exists if the conditions below are satisfied:(H3):

d - γ < E_{2} < d - γ + β - \frac{d}{η}

{(d + β - \frac{d}{η} - E_{2} - γ)}^{2} > \frac{4 d}{η} (E_{2} + γ - d), E_{1} < 1 - x_{2} / η

with γ < d < ηβ or

d < γ < (β - \frac{d}{η} + d)

It is observed that at P₂ & P₃ both the species co-exist under some conditions stated above.

3 Stability analysis

The Jacobian matrix for our model is

J (x, y) = [\begin{array}{c} {ϕ^{'}}_{1} (x) (ϕ_{2} (x) - y) + ϕ_{1} (x) {ϕ^{'}}_{2} (x) & - ϕ_{1} (x) \\ y {ϕ^{'}}_{3} (x) & ϕ_{3} (x) \end{array}] .

(14)

Since $ϕ_{1} (x) = \frac{x}{1 + x},$ therefore ${ϕ^{'}}_{1} (x) = \frac{1}{{(1 + x)}^{2}}$ and from the expressions of ϕ₂(x), ϕ₃(x) we get

\begin{array}{l} {ϕ^{'}}_{2} (x) = 1 - E_{1} - \frac{1}{η} - \frac{2 x}{η}, \\ {ϕ^{'}}_{3} (x) = β {ϕ^{'}}_{1} (x) - \frac{d}{η} = \frac{β}{{(1 + x)}^{2}} - \frac{d}{η} . \end{array}

We first investigate the stability at the trivial equilibrium P₀(0, 0), the boundary equilibrium P₁((1 − E₁)η, 0), and then the discussion of stability of interior equilibrium will be considered in different cases as they appear in terms of their existence.

At P₀(0, 0) the corresponding community matrix is simplified to be

J (P_{0}) = [\begin{array}{c} 1 - E_{1} & 0 \\ 0 & d - E_{2} - γ \end{array}],

whose eigenvalues are (1 − E₁) and (d − E₂ − γ). Therefore the equilibrium at P₀ is stable if

E_{1} > 1 and E_{2} > d - γ

(15)

At P₁((1 − E₁)η, 0) the matrix J is evaluated as

J (P_{1}) = [\begin{array}{c} - (1 - E_{1}) & ϕ_{1} ((1 - E_{1}) η) \\ 0 & ϕ_{3} ((1 - E_{1}) η) \end{array}]

whose eigen values are λ₁ = − (1 − E₁) < 0, since E₁ < 1 for existence of P₁ and λ₂ = ϕ₃((1 − E₁)η).Now λ₂ < 0 if

\frac{β (1 - E_{1}) η}{1 + (1 - E_{1}) η} + d (1 - \frac{(1 - E_{1}) η}{η}) - (E_{2} + γ) > 0

i.e. if

E_{2} > \frac{β (1 - E_{1}) η}{1 + (1 - E_{1}) η} + d E_{1} - γ .

Thus, P₁ is a stable equilibrium if

E_{2} > \frac{β (1 - E_{1}) η}{1 + (1 - E_{1}) η} + d E_{1} - γ

(16)

and E₁ < 1

Now we go through the following cases for discussing the stability at interior equilibrium, taking the references of the cases of existence of equilibrium points discussed earlier.

Case 1.

Let $E_{2} < M i n {(β - \frac{d}{η}) + d - γ, d - γ} and E_{1} < 1 - x_{2} / η,$ then only the positive interior equilibrium is P₂(x₂, ϕ₂(x₂)). At this equilibrium

J = [\begin{array}{c} ϕ_{1} (x_{2}) {ϕ^{'}}_{2} (x_{2}) & - ϕ_{1} (x_{2}) \\ ϕ_{2} (x_{2}) {ϕ^{'}}_{3} (x_{2}) & 0 \end{array}]

The characteristic equation is

λ^{2} - λ ϕ_{1} (x_{2}) {ϕ^{'}}_{2} (x_{2}) + ϕ_{1} (x_{2}) ϕ_{2} (x_{2}) {ϕ^{'}}_{3} (x_{2}) = 0

(17)

Since x₁, x₂ are the roots of ϕ₃(x) = 0, therefore from Eq. (11) we have

ϕ_{3} (x) = - \frac{d}{η} [(x - x_{2}) (x - x_{1})]

where x₂ > x₁ and hence

{ϕ^{'}}_{3} (x) = - \frac{d}{η} [(x - x_{2}) + (x - x_{1})]

so that

{ϕ^{'}}_{3} (x_{2}) = - \frac{d}{η} (x_{2} - x_{1}) < 0

Thus ϕ₁(x₂)ϕ₂(x₂)ϕ₃′(x₂) < 0 as ϕ₁(x₂), ϕ₂(x₂) are positive and hence (17) has one positive and one negative root. Therefore in this case P₂(x₂, ϕ₂(x₂)) is an unstable equilibrium.

Case 2.

Let

(d + β - \frac{d}{η} - E_{2} - γ) < 0 and E_{2} + γ - d < 0 .

In this case P₂(x₂, ϕ₂(x₂)) is the only positive interior equilibrium. As in Case 1, ${ϕ^{'}}_{3} (x_{2}) < 0$ and the equilibrium is unstable at P₂(x₂, ϕ₂(x₂)).

Case 3.

Let

(d + β - \frac{d}{η} - E_{2} - γ) > 0, E_{2} + γ - d > 0

and

{(d + β - \frac{d}{η} - E_{2} - γ)}^{2} > \frac{4 d}{η} (E_{2} + γ - d) .

In this case two interior equilibrium points P₂(x₁, ϕ₂(x₁)) and P₃(x₂, ϕ₂(x₂)) exist.

Now we investigate the stability at P₂ and P₃.

At P₂(x₁, ϕ₂(x₁)) the community matrix is reduced to

J (P_{2}) = [\begin{array}{c} ϕ_{1} (x_{1}) {ϕ^{'}}_{2} (x_{1}) & - ϕ_{1} (x_{1}) \\ ϕ_{2} (x_{1}) {ϕ^{'}}_{3} (x_{1}) & 0 \end{array}]

(18)

whose characteristic equation is

λ^{2} - λ ϕ_{1} (x_{1}) {ϕ^{'}}_{2} (x_{1}) + ϕ_{1} (x_{1}) ϕ_{2} (x_{1}) {ϕ^{'}}_{3} (x_{1}) = 0

(19)

Now $ϕ_{1} (x_{1}) ϕ_{2} (x_{1}) {ϕ^{'}}_{3} (x_{1})$ are all positive under the conditions of existence of P₂(x₁, ϕ₂(x₁)). Thus by Descartes’ rule of signs, P₂ is a node or a focus or a centre. By Routh-Hurwitz condition, the equilibrium at P₂(x₁, ϕ₂(x₁)) is stable if ${ϕ^{'}}_{2} (x_{1}) < 0$ , and is unstable if ${ϕ^{'}}_{2} (x_{1}) > 0$ . A limit cycle is expected closed to the curve ${ϕ^{'}}_{2} (x_{1}) = 0$ . Now ${ϕ^{'}}_{2} (x_{1}) = 0$ implies that $1 - E_{1} - \frac{1}{η} - \frac{2 x_{1}}{η} = 0,$ or $1 - E_{1} = \frac{1}{η} (1 + 2 x_{1}) .$

Or,

E_{1} = 1 - \frac{1}{η} (1 + 2 x_{1}) = 1 - \frac{1}{η} - \frac{1}{d} [(d + β - \frac{d}{η} - E_{2} - γ) - \sqrt{{(d + β - \frac{d}{η} - E_{2} - γ)}^{2} - \frac{4 d}{η} (E_{2} + γ - d)}]

i.e.

E_{1} = ϕ (E_{2})

(20)

where

ϕ (E_{2}) = 1 - \frac{1}{η} - \frac{1}{d} [(d + β - \frac{d}{η} - E_{2} - γ) - \sqrt{{(d + β - \frac{d}{η} - E_{2} - γ)}^{2} - \frac{4 d}{η} (E_{2} + γ - d)]}

(21)

Thus if ${ϕ^{'}}_{2} (x_{1}) < 0$ i.e. if E₁ > ϕ(E₂) then P₂(x₁, ϕ₂(x₁)) is locally asymptotically stable and if E₁ < ϕ(E₂) then P₂(x₁, ϕ₂(x₁)) is unstable in the positive quadrant of x₁x₂ − plane. For E₁ = ϕ(E₂) i.e., for ${ϕ^{'}}_{2} (x_{1}) = 0$ the roots of (19) ur discussion on stability at P₃(x₂, ϕ₂(x₂)) we see that the characteristic equation of the community matrix at P₃ is

λ^{2} - λ ϕ_{1} (x_{2}) {ϕ^{'}}_{2} (x_{2}) + ϕ_{1} (x_{2}) ϕ_{2} (x_{2}) {ϕ^{'}}_{3} (x_{2}) = 0

(22)

Since we have earlier discussed that ${ϕ^{'}}_{3} (x_{2}) < 0, ϕ_{1} (x_{2}) > 0, ϕ_{2} (x_{2}) > 0$ for the existence of P₃; therefore by Descartes’ rule of signs, the roots of (22) are both real and of opposite sign. Thus P₃(x₂, ϕ₂(x₂)) is an unstable saddle point whatever the sign of ${ϕ^{'}}_{2} (x_{2})$ may be.

We now state the following theorem:

Theorem 1

If the conditions (H1), (H2) and (H3) are satisfied then:

(i) P₀(0, 0) is stable if E₁ > 1, E₂ > d − γ.

(ii) P₁((1 − E₁)η, 0) is stable if $E_{2} > \frac{β (1 - E_{1}) η}{1 + (1 - E_{1}) η} + d E_{1} - γ$

and is unstable when $E_{2} < \frac{β (1 - E_{1}) η}{1 + (1 - E_{1}) η} + d E_{1} - γ .$

(iii) P₂(x₁, ϕ₂(x₁)) is stable if E₁ > ϕ(E₂) and is unstable when E₁ < ϕ(E₂).

(iv) P₃(x₂, ϕ₂(x₂)) is always unstable.

We notice that

\frac{d}{d E_{1}} {(T r a c e J)}_{E_{1} = ϕ (E_{2})} = \frac{d}{d E_{1}} {[ϕ_{1} (x_{1}) {ϕ_{2}}^{'} (x_{1})]}_{E_{1} = ϕ (E_{2})} = - \frac{x_{1}}{1 + x_{1}} \neq 0 .

Hence by the Hopf bifurcation theorem (Hassard et al. [15]) the system (9) and (10) enters into a Hopf type small amplitude periodic oscillation at the parametric values E₁ = ϕ(E₂) near the positive interior equilibrium P₂(x₁, ϕ₂(x₁)). Hence we may state the following theorem:

Theorem 2

For E₁ = ϕ(E₂) a super critical Hopf bifurcation takes place for the equilibrium P₂.

Proof

Let us consider the Jacobian matrix (18). By direct calculation the eigen values are given by:

λ_{1,2} = \frac{ϕ_{1} (x_{1}) {ϕ^{'}}_{2} (x_{1}) \pm \sqrt{{ϕ_{1} (x_{1}) {ϕ^{'}}_{2} (x_{1})}^{2} - 4 ϕ_{1} (x_{1}) ϕ_{2} (x_{1}) {ϕ^{'}}_{3} (x_{1})}}{2}

Or, $λ_{1,2} = \frac{T (E_{1}, E_{2}) \pm \sqrt{T^{2} - 4 D (E_{1}, E_{2})}}{2}$ , where T = trace of J and D = det J.

Now for T = 0, D > 0 and in corresponding to the values of E₁ = ϕ(E₂) the matrix has two purely imaginary eigenvalues $λ_{1,2} = \pm i \sqrt{D}$ . Moreover $d / d E_{1} {(Re (λ_{1,2}))}_{E_{1}} = ϕ (E_{2})$ . Thus, all conditions of Hopf theorem are satisfied and a stable limit cycle for E₁ < ϕ(E₂) may be found. All these conditions together prove the existence of a super critical Hopf bifurcation for the equilibrium P₂.

We also have the following result on global stability in the region of local stability:

Theorem 3

If P₂(x₁*, ϕ₂(x₁*)) is locally asymptotically stable, then it is globally stable in the interior of R₂⁺.

Proof

If possible, let there be a periodic orbit Γ = (x(t), y(t)), 0 ≤ t ≤ T with the enclosed region Ω and consider the variational matrix J about the periodic orbit,

J (x, y) = [\begin{array}{c} {ϕ^{'}}_{1} (x) (ϕ_{2} (x) - y) + ϕ_{1} (x) {ϕ^{'}}_{2} (x) & - ϕ_{1} (x) \\ y {ϕ^{'}}_{3} (x) & ϕ_{3} (x) \end{array}] .

We compute

Δ = \int_{0}^{T} {ϕ_{1}}^{'} (x) [ϕ_{2} (x) - y) + ϕ_{1} (x) {ϕ^{'}}_{2} (x) + ϕ_{3} (x)] d t

For biological equilibrium in the region of local stability it can easily be seen that

Δ = \int_{0}^{T} ϕ_{1} (x) {ϕ_{2}}^{'} (x) d t < 0

Therefore, Δ < 0 and the periodic orbit Γ is orbitally asymptotically stable (Cheng et al. [16] and Hale [17]). Since every periodic orbit is orbitally stable then there is a unique limit cycle. From the Poincare–Bendixson Theorem, it is impossible to have unique stable limit cycle enclose a stable equilibrium. This is the desired contradiction. Hence there is no limit cycle and P₂ is globally stable.

As the analytical results already indicate, the existence and stability of the interior equilibrium, as well as the stability of the trivial and boundary equilibrium will depend on the specific parameters that determine the dynamic evolution of the resource stock and the population. In the next section we present a more in-depth analysis of the complexity of the long run dynamics of population and resource stocks as they depend on the parameters of the system. The results are obtained through a detailed numerical and global bifurcation analysis.

4 Numerical simulations

We consider numerical simulations of different cases to understand the theoretical findings.

For η = 60, β = 4, d = 0.6, γ = 0.9 we have the bifurcation diagram shown in Fig. 1, and the point (E₁, E₂) = (0.9, 2.69) marked by * is in stable region and the point (E₁, E₂) = (0.75, 2.69) marked by • is in the region of unstability.

Fig. 1
Bifurcation diagram in the (E₁, E₂) plane.

Fig. 1 illustrates a detailed picture of the possible long run dynamics as they depend on the values of the harvesting efforts E₁ and E₂ for the benchmark case. The various dynamic behaviors are described in separate figures that show the corresponding stable and unstable equilibria in phase space together with selected time paths.

In brief, the bifurcation diagram as presented in Fig. 1 allows us to investigate the specific combination of efforts E₁ and E₂ that may cause the model to shift from sustainable long run equilibrium towards a limit cycle and finally the collapse of the system. Local bifurcation theory is the numerical tool that determines the boundaries between these regions in parameter space (Fig. 2).

We see that as d increases from 0.6 to 0.9 the area of region of stability decreases as shown in Fig. 2. If we increase d again to 1.0 the stable region of stability shrinks more, as depicted in the Fig. 3.

Figs. 1, 2 and 3 show bifurcation diagrams in the (E₁, E₂)-plane for d = 0.6, 0.9 and 1, respectively. We see that the stability region becomes smaller as d becomes larger. We may say that d destabilizes the coexisting equilibrium P₂. Thus for the long run sustainability of a prey-predator system, this might require reducing alternative prey (Fig. 4).

Fig. 4
The phase diagram for (E₁, E₂) = (0.9, 2.69) with d = 0.6.

The phase diagram for d = 0.06 is shown in Fig. 4. When d increases to 0.9 or 1.0 then the point (E₁, E₂) = (0.9,2.69) marked by * lies outside the stable region and the interior equilibrium becomes unstable. The corresponding phase diagrams are shown in the Figs. 5 and 6.

Fig. 5
The phase diagram for (E₁, E₂) = (0.9, 2.69) with d = 0.9.

Fig. 6
The phase diagram for (E₁, E₂) = (0.9, 2.69) with d = 1.0.

Figs. 7 and 8 shows the phase diagram of prey and predator with a fixed value of E₂ = 2.69 and different values of E₁. Fig. 9 has E₁ = 0.39, and E₂ = 3.60

Fig. 7
The phase diagram for (E₁, E₂) = (0.75, 2.69) from region-II of the Fig. 1.

Fig. 8
The phase diagram for (E₁, E₂) = (2, 2.69) in region-III of Fig. 1.

Fig. 9
The phase diagram for (E₁, E₂) = (0.39, 3.60) in the Region IV of Fig. 1.

Figs. 10 and 11 shows the isoclines of prey and predator with a fixed value of E₂ = 2.69 and different values of E₁. It is observed that the equilibrium point for the prey population gradually decreases when the respective fishing effort used to harvest prey population is simultaneously increased. In Fig 10, d = 0.6 and in Fig. 11 d = 0 (Figs. 12 and 13).

Figs. 12 and 13 show the isoclines of prey and predator population with a fixed value of E₁ = 0.6 and different values of E₂. It is observed that the equilibrium point for the prey population gradually increases as the effort E₂ increases. It is natural as an increase in E₂ decreases the predator population and hence enhancing the survival rate of the prey.

5 Conclusion

Managing predator-prey systems involves complex challenges for resource managers. Many studies have demonstrated populations consequences of predators on prey population, but how managers should use this information are not easy to decide. Predator control can be effective at enhancing survival and recruitment in populations of prey, but the potential role of predators in structuring ecological communities has been recognized for sometime. For example, overexploitation of sea otter populations resulted in increase in sea urchins that subsequently destroyed kelp beds that provided habitats for fish and other marine organisms. Because of such complexity of food webs in ecological communities it is difficult to anticipate the full ramifications of eliminating or restoring predators. For example, reducing predator numbers to increase abundance of prey can have counter-productive results such as increasing disease and parasite infection in prey.

We believe that prey-predator management requires ecosystem management and this must include careful consideration of habitats as well as the particular predator prey populations. Management actions include adjusting season length and bag limits for hunter and trappers, but must also include other activities such as management of habitats that provide secure areas for prey. Ecosystem management acknowledges the value of predators in the environment.

Thus unregulated exploitation and extinction of many natural and biological resources is a major problem of present day. This work pays attention to the exploitation or harvesting of such resources. It describes some strategies on harvesting efforts of a prey-predator model incorporating an alternative prey for the predators. It gives a study of a prey-predator model with an alternative prey, which shows some methods to control the system and how the state can be driven into equilibrium. We have considered all possible cases for the variation of parameters for the equilibrium and stability analysis of the model. It has also been discussed, how the system bifurcates from equilibrium. We have seen that the interior equilibrium in Case 3 is stable for E₁ > ϕ(E₂) and is unstable for E₁ < ϕ(E₂) and another interior equilibrium is always unstable. Some numerical examples also show how the system becomes stable at an interior equilibrium and how it bifurcates from the equilibrium and also some other features of the model.

Managing prey-predator populations for hunter harvest becomes more complex when predators are competing with humans for the same prey. Adjustments to harvest regimes may be necessary, but certainly we can have sustainable harvest of populations under predation. Elk on the northern range of Yellowstone National Park are harvested by hunters when they move into Montana during winter. The sustainability of this harvest is ensured because the Montana Department of Fish, Wildlife and Parks has density dependent harvest guidelines so that the number of tags issued for the late-Gardiner elk hunt increases with the number of elk censused on the northern range. This helps to balance the hunter harvest with wolf predation ensuring that the elk population is not driven to low levels by excessive hunter harvest.

Model that has been developed permit harvest guidelines while accommodating predators and these can be used to achieve sustainable yields. However application of this model will require data to manage a prey-predator systems.

^☆ Research of T.K. Kar was supported by the Council of Scientific and Industrial Research (CSIR), India (Grant No. 25 (0160)/08/EMR-II dated: 17.01.08) and research of S.K. Chattopadhyay was supported by the University Grants Commissions (UGC), India (Grant No. PSW-179/09-10(ERO) dated: 08.10.2009).