EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 2, 2024, 1294-1305 ISSN 1307-5543 – ejpam.com Published by New York Business Global Determination of the Fixed Point of Lotka-Volterra Function Mary Osei Fokuo1, William Obeng-Denteh1, Isaac Kwame Dontwi1, Patrick Akwasi Anamuah Mensah2,∗ 1 Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana. 2 Department of Mathematics and ICT, St Ambrose College of Education, Dormaa Akwamu, Ghana Abstract. The paper focuses on the Lotka-Volterra function in its discrete form. The purpose of the study was to determine the fixed points of the function. The study employs the Banach Fixed Point Theorem and Contraction Mapping in Metric Space on the function to demonstrate the uniqueness of the fixed points and its continuous stability after several iterations, using the fixed points as the initial conditions. The study has shown that (0, 0), ( 0, α−1 β ) , ( 1+γ δ , 0 ) and( 1+γ δ , α−1 β ) are the fixed points of the function, with the initial pair serving as a trivial one and the other three solely depending on the parameter values for the behavior of the function. The outcome of the limit points of the function as the fixed points after several iterations forms a fixed orbit structure of the function, irrespective of the value of the parameter. The study also showed the uniqueness of the fixed points, demonstrating the stability and continuity of the function in its steady state. 2020 Mathematics Subject Classifications: 54H25, 47H10, 54E35 Key Words and Phrases: Lotka-Volterra, fixed point, parameter, solution, function,contraction mapping, fixed point theorem 1. Introduction Many researchers have studied about fixed point theorem. All their definitions seem to have one idea. That is fixed point theorem is where each fixed point of a function G must exist, x ∈ X such that G(x) = x. [9] wrote a brief historical survey on fixed point theorem. Many papers were cited in that paper. Many interesting results on fixed point theorem were also given like generalization and extension of fixed point theorem through ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5155 Email addresses: abenamof@gmail.com (M. O. Fokuo), wobengdenteh@gmail.com (W. Obeng-Denteh), ikdontwi@knust.edu.gh (I. K. Dontwi), patrickakwasianamuahmensah@sace.edu.gh (P. A. A. Mensah) https://www.ejpam.com 1294 © 2024 EJPAM All rights reserved. M. O. Fokuoet al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1294-1305 1295 different types of mappings. Suppose T represent a self-map on set X. A fixed point of the mapping T is referred to as an element x in X such that Tx = x. One of the most important theorems in fixed point theorem is the L.E. Brouwer’s fixed point theorem in which it is said that each continual self-mapping of the sealed unit in the n− dimensional Euclidean spaces Rn, possess a fixed point[5]. Advanced fixed point theorem for internal mapping by employing a known Ky Fan type outcome in a Hilbert space setting.[10] For two species (prey and predators) to exist, there is an equation which model the struggle. This model was brought up by two scientists: Lotka and Volterra.The Scientists came to a conclusion based on the problem they had in 1920 by Lotka and 1926 by Volterra. The conclusion was the same, that the interaction of the two species would give rise to periodic Oscillation in their populations.[2] Many works have been done on the fixed-point theorem and the Lotka Volterra function, but our focus is on the determination of the fixed point of the Lotka Volterra function, applying the Banach fixed point theorem, and the contraction mapping on the Lotka Volterra function to find the fixed point of the Lotka Volterra function. 2. Preliminaries Definition 1. A fixed point is a point that remains the same after applying a map system of differential equations, etc. A point x0 is referred to as a function’s fixed point, if g (x0) = x0 , [11]. Definition 2. The fixed point theorem in general term is stated as an outcome of a function having at least a fixed point under a certain condition [11] Definition 3. Let U be a non-empty set. Then, the real function d (distance function) that assign any ordered pair d(u, v) of element u, v and w ∈ U is a metric space, if the following properties are satisfied: 1. d(u, v) ≥ 0 and d(u, v) = 0 if and only if u = v 2. d(u, v) = d(v, u) 3. d(u, v) + d(v, w) ≥ d(u,w). A function d satisfying the conditions (1)− (3) is a metric on U [8] Definition 4. A dynamic system is one in which a function explains how a point’s re- lationship to time changes with a given environment. Examples are the mathematical formulas that explain how water moves through a pipe, how many fish spawn in a lake each springtime etc.[4] Definition 5. A steady state of ẋ = f(x) is a point x ∈ U as to which f(x) = 0. A steady state x∗ is said to be Lyapunov stable if for any ε > 0, there exist δ > 0 this way for all x0 with |x∗ − x0| < δ we have |φ (x0, t)− x∗| < ε for all t ≥ 0, [3]. Definition 6. An equilibrium x is said to be globally asymptotically stable in the set of all positive solutions.[7] M. O. Fokuoet al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1294-1305 1296 Definition 7 (Lotka-Volterra Equations). Xn+1 = αxn − βxnyn (1) Yn+1 = δxnyn − γyn (2) Where n = 0, 1, 2, . . . , x is the quantity of preys, y is the quantity of predators, xn+1 and yn+1 are the two population growth rates and α, β, δ, γ are positive real parameters defining the interaction between the two species [2]. Definition 8 (Population Equilibrium). The model reaches population equilibrium when none of the population levels is shifting. When both derivatives equal 0, that is when it occurs. xn (α− βyn) = 0 (3) yn (γ − δxn) = 0 (4) Hence, there are two solutions to the equations or the systems. {yn = 0, xn = 0} and { yn = α β xn = γ δ } , Hence, there are two equilibria.[7] Theorem 1 (Banach Contraction Principle). Suppose(P, d) is a complete metric space and H : P → P is a mapping of contractions using the Lipschitz constant k < 1. Then, the fixed point ω ∈ P , for all x ∈ P is a unique point in H. That is; limn+∞Hn(x) = ω.Moreover, for each x ∈ P, we have d(Hn(x), ω) ≤ kn 1−kd(H(x), x) . [1] Theorem 2. Given (P, d) as a complete metric space, and a mapping H : P → P for which HN is a contraction mapping for N ≥ 1. As a result, H has a distinct fixed point. In general, it is unclear if H has a fixed point whenever HN has a fixed point. The term “periodic points of H” also applies to fixed point of HN .[1]. 3. Main Work Under this part we look for the main solutions of the Lotka Volterra function and also determine the fixed point of the function. 3.1. The zeros of the Lotka Volterra function In this section, we look for the roots, or zeros, of the function. In determining the zeros or roots of the function, let Xn+1 = 0, from equation (1), thus Xn+1 = αxn−βxnyn becomes αxn − xnβyn = 0 (5) xn(α− βyn) = 0 (6) It implies that xn = 0 and α− βyn = 0 then yn = α β Hence, the root of Xn+1 = αxn − βxnyn thus (xn, yn) is ( 0, αβ ) M. O. Fokuoet al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1294-1305 1297 Also, let yn+1 = 0 then from equation (2) thus, Yn+1 = δxnyn − γyn also becomes δxnyn − γyn = 0 (7) yn(δxn − γ) = 0 (8) Then yn = 0 and δxn − γ = 0 implies xn = γ δ Similarly, the root of Yn+1 = δxnyn − γyn thus (xn, yn) is (γ δ , 0 ) Therefore, the roots are (0, 0) , ( 0, αβ ) , (γ δ , 0 ) , and ( γ δ , α β ) 3.2. The solutions of the Lotka Volterra function This section is mainly about the solutions of the function. Let Xn+1 = xn (9) Yn+1 = yn (10) Equating equation (1) and equation (9) becomes αxn − xnβyn = xn αxn − xnβyn − xn = 0 implies xn(α− βyn − 1) = 0 Then xn = 0 and α− βyn − 1 = 0 βyn = α− 1 Therefore, yn = α−1 β Also equating equation (2) and equation (10) becomes δxnyn − γyn = yn (11) Then δxnyn − γyn − yn = 0 yn(δxn − γ − 1) = 0 Then yn = 0 and δxn − γ − 1 = 0 δxn = 1 + γ therefore, xn = 1+γ δ Hence, the solutions of the function are ( 0, α−1 β ) and ( 1+γ δ , 0 ) 3.3. Determination of the fixed point of the Lotka Volterra function In this section, we will consider the two definitions of the function. That is; xn+1 = αxn − xnβyn and yn+1 = δxnyn − γyn and then work out for the fixed point of the function using the solutions, (0, 0), ( 0, α−1 β ) , ( 1+γ δ , 0 ) , and ( 1+γ δ , α−1 β ) . We then apply the idea and the definition of the theorem of fixed points and fixed point. That is; a function G is fixed point theorem having a minimum of one fixed point x ∈ X such that G(x) = x. A point x0 is the fixed point of a function g(x), such that g (x0) = x0. M. O. Fokuoet al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1294-1305 1298 1) Determining the fixed point of Xn+1 = αxn − xnβyn At (0, 0) Xn+1 = α(0)− (0)β(0) Implies Xn+1 = 0− 0 Xn+1 = 0 Hence, (0, 0) is one of the fixed points but trivial. Also, at( 1+γ δ , α−1 β ) Xn+1 = α ( 1+γ δ ) − ( 1+γ δ ) β ( α−1 β ) implies Xn+1 = (α+αγ δ ) − ( 1+γ δ ) (α− 1) = (α+αγ δ ) − [(α+αγ δ ) − ( 1+γ δ )] = (α+αγ δ ) − (α+αγ δ ) + ( 1+γ δ ) = ( 1+γ δ ) Again, ( 1+γ δ , α−1 β ) is also a fixed point but its existence will depend on the values of the parameter of the function. 2) Determining the fixed point of Yn+1 = δxnyn − γyn Then at (0, 0) Yn+1 = δ(0)(0)− γ(0) Yn+1 = 0− 0 Yn+1 = 0 Also, at ( 1+γ δ , α−1 β ) Yn+1 = δ ( 1+γ δ )( α−1 β ) − γ ( α−1 β ) implies Yn+1 = (1 + γ) ( α−1 β ) − γ ( α−1 β ) = (1 + γ) ( α−1 β ) − γ ( α−1 β ) = ( α−1 β ) + ( γα−γ β ) − ( γα−γ β ) = ( α−1 β ) Hence, (0, 0), ( 0, α−1 β ) , ( 1+γ δ , 0 ) and ( 1+γ δ , α−1 β ) are the fixed points of the function. Final Results In this section we impose the contraction mapping on the Lotka Volterra to see the outcome of its behaviour[6]. Definition 9 (Contraction Mapping in Metric Space). Given (M,d) a metric space, a function T : M → M is said to be a contraction mapping if there is a constant a constant M. O. Fokuoet al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1294-1305 1299 q with q < 1 such that for all x, y ∈ M d(T (x), T (y)) ≤ q · d(x, y) Applying the Banach fixed point theorem and the contraction mapping on the Lotka Volterra functions. Using Banach fixed point theorem to find the fixed point of Lotka Volterra functions. From the definition of Banach fixed theorem, let (M, d) be a complete metric space then every contraction has a unique fixed point. If T (x) = x, T (y) = y then d(x, y) = d(T (x), T (y)) ≤ q · d(x, y) q < 1 so d(x, y) = 0 or x = y To show that a fixed point exists, pick any x ∈ M . Setting x ∈ x0, we define a sequence {xi}i∈z+ by setting xn+1 = T (xn) xn+1 = αT (xn)−βT (xn) (yn) xn+1 = (α−β(yn))T (xn) Rewriting the contraction formula we have xn+1 = (α− β(yn))T (xn) d(xn+2, xn+1) ≤ (α− β(yn))qd(xn+1, xn) d(xn+2, xn+1) ≤ (α− βyn)qd(xn+1, xn) d (xn+1, xn) ≤ (α− βyn) q nd (x1, x0) d (xn+1, xn) ≤ αqnd (x1, x0)− βqnynd (x, x0) ≤ qn[αd (x1, x0)− βynd (x1, x0)] d(xn+1, xn) ≤ qn[(α− β)ynd (x1, x0)] Assuming n < m d (xm, xn) ≤ d (xm, xm−1) + d (xm−1, xm−2) + . . .+ (xn+1, xn) d (xm, xn) ≤ ( qm−n−1 + qm−n−2 + . . .+ q + 1 ) d (xn+1, xn) ≤ ( 1− qm−n 1− q ) d (xn+1, xn) ≤ ( 1− qm−n 1− q ) qn (α− βyn) d (x1, x0) since qm=n < 1 d(xm, xn) < qn 1− q (α− βyn)d(x1, x0) Thus {xi} is Cauchy. This shows that (xn) is Cauchy sequence in X. Hence, (xn) must be convergent, say lim n→+∞ xn = x M. O. Fokuoet al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1294-1305 1300 Since T is continuous, we have Tx = T ( lim n→+∞ xn ) = lim n→+∞ T (xn) = lim n→+∞ xn+1 Since the limit of xn+1 is the same as that of (xn) Thus, x is a fixed point of T. ILLUSTRATION 1 Xn+1 = g(xn, yn) = αxn − xnβyn Then, by considering the coordinate of y, that is yn = α− 1 β lim yn→α−1 β g(xn, yn) = lim yn→α−1 β β(αxn − xnβyn) = xn lim yn→α−1 β (α− βyn) where n = 0, 1, 2, . . . implies lim yn→α−1 β g(xn, yn) = xn [ lim yn→α−1 β (α)− lim yn→α−1 β βyn ] = xn[α− β(α− 1)] = xn × 1 = xn Hence, lim yn→α−1 β Xn+1 = lim yn→α−1 β g(xn) = xn irrespective of the values of the parameters. This implies: when n = 0, x1 = g(x0) = x0. When n = 1, X2 = g(x1) = x1. When n = 2, X3 = g(x2) = x2. Hence, the orbit {x0, x1 = g(x0) = x0, x2 = g(x1) = x1, x3 = g(x2) = x2, . . .} Serve as the fixed points of the function since it is a repeated point that is unique and asymptotically stable throughout the iteration process. Example 1. Given f(xn, yn) = αxn − xnβyn. Let γ > 1, δ > 1, α > 1, and β > 1. At γ = 1.1, δ = 2.1, α = 1.2, and β = 2.5.( 1 + γ δ , α− 1 β ) implies (xn, yn) Then, lim yn→α−1 β f(xn, yn) = lim yn→α−1 β (αxn − xnβyn) M. O. Fokuoet al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1294-1305 1301 At α = 1.2, β = 2.5 Implies lim yn→α−1 β f(xn, yn) = xn lim yn→α−1 β (1.2− 2.5yn) When n = 0, x0 = 1 + γ δ = 1 + 1.1 2.1 = 1, y0 = α− 1 β = 1.2− 1 2.5 = 0.08 Implies (x0, y0) = (1, 0.08) impliesX1 = lim yn→α−1 β f(x0, y0) = lim y0→0.08 f(x0, y0) = lim y0→0.08 f(1, 0.08) = x0 lim y0→0.08 (1.2− 2.5y0) = 1[1.2− 2.5(0.08)] This implies X1 = x0 = 1 Now for X2, we iterate the function again using x1 = 1 and use different values for the parameters for y coordinate, that is;α = 10, β = 12 at n = 1 That is lim y1→α−1 β f(x1, y1) = x1 lim y→ α−1 β (10− 12y1). where y1 = 10− 1 12 = 9 12 = 0.75. Hence (x1, y1) = (1, 0.75) Therefore X2 = lim yn→α−1 β f(x,y1) = lim y1→0.75 f(x1, y1). = x1 lim y1→0.75 (10− 12y1) = 1[10− 12(0.75)] This implies X2 = x1 = 1 The iteration process so far indicates the limit of xn+1 is the same as that of (xn) and keeps repeating itself. Hence, xn = 1 + γ δ is a fixed point of the function. Which implies that when n = 0, X1 = g(x0) = x0 when n = 1, X2 = g(x1) = x1 when n = 2, X3 = g(x2) = x2 ... when n = k,Xk+1 = g(xk) = xk Considering equation (12), that is yn+1 = δxnT (yn)− γT (yn) M. O. Fokuoet al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1294-1305 1302 We pick any y ∈ M setting y = y0, we define a sequence {yi}i∈Z+ by setting yn+1 = T (yn) yn+1 = δxnT (yn)− γT (yn) Rewriting the contraction formula we have yn+1 = (δxn − γ)T (yn) d(yn+2, yn+1) ≤ (δxn − γ)qd(yn+1, yn) or d(yn+1, yn) ≤ (δxn − γ)qnd(y1, y0) d(yn+1, yn) ≤ qn(δxn − γ)d(y1, y0) Assuming n < m d(ym, yn) ≤ d(ym, ym−1) + d(ym−1, ym−2) + . . .+ d(yn+1, yn) d(ym, yn) ≤ (qm−n−1 + qm−n−2 + . . .+ q + 1)d(yn+1, yn) d(ym, yn) ≤ 1− qm−n 1− q d(yn+1, yn) since qm−n < 1 d(ym, yn) < qn 1− q (δxn − γ)d(y1, y0) Thus {yi} is Cauchy. This shows that yn is a Cauchy sequence in M Hence, (yn) must be convergent, say lim n→+∞ yn = y Since T is continuous, we have Ty = T ( lim n→+∞ yn ) = lim n→+∞ Tyn = lim n→+∞ yn+1 = y Since the limit of yn+1 is the same as that of yn Thus, y is a fixed point of T. ILLUSTRATION 2 Similarly, let Yn+1 = h(xn, yn) = δxnyn − γyn Then, by considering the coordinate of x, that is xn = 1 + γ δ lim xn→ 1+γ δ h(xn, yn) = lim xn→ 1+γ δ (δxnyn − γyn) = yn lim xn→ 1+γ δ (δxn − γ) where n = 0, 1, 2, . . . = δ ( 1 + γ δ ) yn − γyn M. O. Fokuoet al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1294-1305 1303 = yn [ δ ( 1 + γ δ ) − γ ] = yn[(1 + γ)− γ] = yn × (1) = yn Hence, Yn+1 = h(yn) = yn irrespective of the values of the parameters. whitewhitespaceere when n = 0, Y1 = h(y0) = y0 when n = 1, Y2 = h(y1) = y1 when n = 2, Y3 = h(y2) = y2 This forms the orbit {y0, y1 = h(y0) = y0, y2 = h(y1) = y1, y3 = h(y2) = y2, . . .} of the function that are equilibrium in nature, asymptotically stable and continuous after several iterations. Example 2. Given Yn+1 = h(xn, yn) = δxnyn − γyn. Let γ > 1, δ > 1, and α > 1, β > 1. At γ = 1.1, δ = 2.1, α = 1.2, β = 2.5.( 1 + γ δ , α− 1 β ) implies (xn, yn)) Then, taking the limit of the function Yn+1as xn → 1 + γ δ for γ = 1.1, δ = 2.1 implies lim xn→ 1+γ δ h(xn, yn) = lim xn→ 1+γ δ (δxnyn − γyn) lim xn→ 1+γ δ h(xn, yn) = lim xn→ 1+γ δ (2.1xnyn − 1.1yn) = yn lim xn→ 1+γ δ (2.1xn − 1.1) Then for xn = 1 + γ δ ,When n = 0, implies x0 = 1 + γ δ = 1 + 1.1 2.1 = 1, Also, for y1 implies Y1 = lim xo→1 h(x0, y0) = y0 lim x0→1 (2.1x0 − 1.1) = 0.008[2.1(1)− 1.1] = 0.08 Hence, y1 = y0 = 0.08 Now, using y1 = 0.08 as the initial value for the next iteration Y2 and taking different values for the parameters of x coordinate, that is; γ = 10, δ = 12 at n = 1 That is lim xn→ 1+γ δ h(xn, yn) = lim xn→ 1+γ δ (12xnyn − 10yn) M. O. Fokuoet al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1294-1305 1304 = yn lim xn→ 1+γ δ (12xn − 10) x1 = 11 12 = 0.92 (x1, y1) = (0.92, 0.08) Y2 = lim x1→0.92 h(x1, y1) = y1 lim x1→0.92 (12x1 − 10) = 0.08[12(0.92)− 10] = 0.08 Again the final value of y2 = y1 = 0.08 Hence, from the iteration process so far, the limit of Yn+1 is the same as that of (yn) and keeps repeating itself. Thus,indicating that yn = α−1 β is a fixed point of the function. Which implies that: when n = 0, Y1 = h(y0) = y0 when n = 1, Y2 = h(y1) = y1 when n = 2, Y3 = h(y2) = y2 ... when n = k, Yk+1 = h(yk) = yk Clearly, at a fixed value of γ, δ, α, β for h(xn, yn) and f(xn, yn) , i.e., (γ = 1.1, δ = 2.1, α = 1.2, β = 2.5),The functions have fixed values, for instance, (1, 0.08) irrespective of the number of successive iterations and the values for the parameters of the x and y coordinates. This indicates that the structure of the fixed orbits of the function is in equilibrium as it travels through time with a stable and continuous movement. 4. Conclusion The Lotka-Volterra function has been studied, and it shows that the function has two sets of roots, or zeros. (0, 0) and ( γ δ , α β ) where the latter depends on the parameters of the function. Again, there are four solutions of the function (0, 0), ( 0, α−1 β ) , ( 1+γ δ , 0 ) and( 1+γ δ , α−1 β ) , where (0, 0) as a trivial solution always exits but the existence of ( 1+γ δ , α−1 β ) depends on the parameters of the function.The study also shows that the solutions of the function are the fixed points of the function, and the limit points, as the fixed points of the function, are asymptotically stable and continuous after several iterations. The outcome of the fixed points after several iterations forms a fixed orbit structure of the function, irrespective of the value of the parameter. In addition, the uniqueness of the fixed points demonstrates the stability and continuity of the function in its steady state. 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