Optimal Harvesting Policy for the Beverton–Holt Quantum Difference Model

In this paper, we introduce exploitation to the Beverton– Holt equation in the quantum calculus time setting. We first give a brief introduction to quantum calculus and to the Beverton–Holt q-difference equation before formulating the harvested Beverton–Holt q-difference equation. Under the assumption of a periodic carrying capacity and periodic inherent growth rate, we derive its unique periodic solution, which globally attracts all solutions. We further derive the optimal harvest effort for the Beverton–Holt q-difference equation under the catch-per-effort hypothesis. Examples are provided and discussed in the last section.


Introduction
Beverton and Holt introduced their population model in the context of fisheries [3] in 1957 as (1) x n+1 = νKx n K + (ν − 1)x n , n ∈ N 0 , where x 0 > 0, ν > 1 is the inherent growth rate, and K > 0 is the carrying capacity.
The model is applied in various fields such as biology, economy and social science, see [2,3,18,15]. To achieve a more realistic presentation of population dynamics, additional assumptions have been added to the traditional model such as contest competition [12], within-year resource limited competition [14], and survivor-rates [13]. In [17], the authors considered modifications of the Beverton-Holt model and the authors of [16] discussed the sigmoid Beverton-Holt model.
In [11], the authors investigated (1) on time scales. Recently, assuming a periodically forced environment and periodic growth rate, (1) was analyzed and the Cushing-Henson conjectures for the case of periodic coefficients were discussed in [7].
In [10], the authors discussed the Beverton-Holt equation with exploitation that reads as (2) (1 + h n )x n+1 = K n ν n x n K n + (ν n − 1)x n , where h represents the harvest effort.
The following theorems were proved in [10].
Theorem 1.2 (See [10]). Assume (3) and (in order to guarantee a nonnegative harvest effort) The optimal harvest effort for (2) is and the maximal harvest yield over one period is In this paper, we include exploitation to the periodically forced Beverton-Holt equation in the quantum calculus setting, which is classically defined as where x 0 > 0, and ν, K : q N 0 → R are the inherent growth rate and carrying capacity, respectively. In [4], the authors analyzed the solution of classical quantum Beverton-Holt model for one-periodic growth rate ν and also discussed the Cushing-Henson conjectures for the case of a one-periodic inherent growth rate. The case of a one-periodic inherent growth rate for the q-difference equation corresponds to a constant inherent growth rate in the classical Beverton-Holt differential/difference equation. In [8,9], the Beverton-Holt q-difference equation, assuming periodic growth rate and periodic carrying capacity, as investigated and formulations related to the Cushing-Henson conjectures were presented. In this work, we continue the discussion of the Beverton-Holt q-difference equation from an economical perspective by including exploitation by a catch-per-effort hypothesis. We formulate the model and derive its periodic solution, which is shown to be globally asymptotically stable. Further, the maximum sustainable yield for the harvested Beverton-Holt q-difference equation is derived.

Some Quantum Calculus Essentials
In this section, we provide some quantum calculus prerequisites. Throughout, let q > 1.
The set of all regressive functions is denoted by R. Moreover, p ∈ R is called positively regressive, denoted by p ∈ R + , if Using the introduced function µ, the derivative can be defined as follows.
Definition 2.3. The derivative of a function f : q N 0 → R is given by Definition 2.4 (See [4]). Let p ∈ R and s ∈ q N 0 . The exponential function is defined by e p (s, s) = 1, and e p (t, s) = 1 ep(s,t) for t < s. It is not hard to show that the following property holds.
Theorem 2.1. If p ∈ R, then e p (t, s) = e p (t, r)e p (r, s) for all s, t, r ∈ q N 0 . Theorem 2.2 (See [5, Theorem 2.44]). If p ∈ R + and t 0 ∈ q N 0 , then e p (t, t 0 ) > 0 for all t ∈ q N 0 . The integral in quantum calculus is defined in the following way.
In particular in the last section, the following operations will be used. Definition 2.6 (See [6, p. 10]). Define the "circle plus" addition on R as and the "circle minus" subtraction as e p (t, s) = e p (s, t) = 1 e p (t, s) .
Besides the circle plus and circle minus operation, a circle dot operation is defined. [6, p. 18]). The circle dot multiplication of a constant value α and a function p ∈ R + is defined as Example 2.1. Let p ∈ R + and α = 1 2 . Then Note that by the definition of the dot multiplication, We furthermore need the definition of periodicity for functions f : q N 0 → R.

The Beverton-Holt q-difference equation with exploitation
The Beverton-Holt q-difference equation was presented in [4] as is the intrinsic growth rate, and x : q N 0 → R + represents the population density.
Using the substitution α = v−1 µv , we obtain the logistic dynamic equation that is well studied in [5]. We introduce exploitation to (8) by the catch-per-effort hypothesis, which yields where H : q N 0 → R + represents the harvest effort. When studying H(t), we should be aware that the time intervals in the quantum calculus setting are increasing. We can therefore express H more explicitly as This allows us to investigate the harvest effort reduced by the time stretching property.
Applying the substitution α = v−1 µv to (9), we obtain which is equivalent to Optimal Harvesting Policy i.e., i.e., where Recall that the logistic differential equation including exploitation is given in the similar form The q-difference equation (11) is solved by using the transformation u = 1/x, which yields This is in the form of a first-order q-difference equation with the solution given in [5] by 3.1. Existence and uniqueness Theorem. In this section, we are interested in providing conditions for the existence and uniqueness of a periodic solution. We aim to prove the following theorem.
Then (10) has a unique ω-periodic solution which globally attracts all positive solutions.
Let us first present the following lemmas that will assist us in the analysis.
Proof. We have which completes the proof since f g = f ⊕ ( g).
Proof. Let a, b ∈ N 0 such that t 0 = q a and t = q b , and assume w.l.o.g.
For the second equation, note that which completes the proof. Proof. We have since µα ∈ (0, 1).

The optimal sustainable yield
In order to discuss the maximum sustainable yield, let us recall that in quantum calculus, the time steps increase as time passes. To take this change of time intervals into consideration, we analyze the average of the harvest at each time step. This yields the formulation of the average of the sustainable yield Theorem 4.1. Assume (14). Then the sustainable yield over one period is maximal for Remark 4.1. The harvest yield has the property for any t * ∈ q N 0 .
In order to prove Theorem 4.1, the following lemmas will be useful.
Proof. Assume first t > s. Then If t < s, then and if t = s, then e f ∆ /f (t, s) = 1.
which completes the proof. Proof. Let i, n ∈ N 0 such that t = q i and s = q n . Then for t > s e ql σ (t, s) = For t < s, e ql σ (t, s) = 1 e ql σ (s, t) = 1 e l (qs, qt) = e l (qt, qs) and if t = s, then e ql σ (t, s) = 1 = e l (qt, qs).
which completes the proof.
Note that h * is a one-periodic function, i.e., where α = a t . Figure 1 shows the optimal harvest effort h * and also h * without the time-stretching factor, i.e., h * t = H for α = a t and a = 0.3. Figure 1. The optimal harvest h * (stars) and h * t = H (dots). Figure 2 shows the relation of h * with respect to the growth rate α reduced by its time-stretching character, i.e., αt = a. Note that this is a similar behavior as in the case T = Z with constant coefficients, where the optimal harvest effort h * had the behavior with respect to the one-periodic/constant growth rate as visualized in Figure 3. The difference in the values is caused by the fact that h * is expressed without the time-stretching factor.