EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5643 ISSN 1307-5543 – ejpam.com Published by New York Business Global Dynamics of Deposit and Loan Volumes Model with Non-Performing Loan and Deposit Withdrawal Factors Onik Febria Damayanti1, Isnani Darti1,∗, Agus Suryanto1 1 Department of Mathematics, Faculty of Mathematics and Natural Sciences, Brawijaya University, Malang, East Java, Indonesia Abstract. Non-performing loan (NPL) and deposit withdrawal are two factors that can affect loan and deposit volumes in the bank. The distribution of deposits into loans at banks can be modelled using the predator-prey model. In this research, we propose and analyse a mathematical model dealing with two loans, i.e. a loan for individuals and a loan for companies at one bank, which is developed from two predators and one prey model. We aim to study the dynamics and long-term behaviour of the proposed model, as well as to discuss the effects of the NPL and deposit withdrawal parameters associated with the model. The results of the analysis show that the model has five equilibrium points. We find that the equilibrium point without deposit and loan activities in the bank is always unstable, while the other equilibrium points are globally asymptotically stable if their certain conditions are satisfied. The theoretical results are verified by our numerical simulations. 2020 Mathematics Subject Classifications: 91B74, 37N40, 34C25, 92D25 Key Words and Phrases: Deposit and Loan Model, Non-Performing Loan, Withdrawal, Predator- Prey, Global Stability 1. Introduction A bank is a business entity that collects deposits from the public and then lends them back to the public in the form of loans, otherwise known as credit. In Indonesia, OJK or the Financial Services Authority classifies commercial banks into four KBMI or Bank Groups based on Core Capital. Core capital is a component of capital sourced from the bank itself that can affect the bank’s operations in a stable and safe manner, so the amount of core capital of a bank affects the management of deposits from the public [5, 12, 26]. Thus, the KBMI classification causes banks to have their limit on receiving deposits from the public. A bank will reward people who deposit their funds with interest on their deposits. The volume of deposits will be distributed into loans by adding loan interest ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5643 Email addresses: isnanidarti@ub.ac.id (I. Darti), onikfeda@student.ub.ac.id (O. F. Damayanti), suryanto@ub.ac.id (A. Suryanto) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) O. F. Damayanti, I. Darti, A. Suryanto / Eur. J. Pure Appl. Math, 18 (1) (2025), 5643 2 of 24 [12]. The distribution of deposit volumes into loan volumes is a way for banks to obtain funds for operational costs and return deposit interest to their customers [27]. People who deposit their funds in the bank can also withdraw their deposits in cash or transfer them to another account according to the bank’s requirements [10, 12, 14]. Thus, the bank’s ability to meet withdrawal requests and other cash outflows serves as a crucial indicator to assess its financial health and overall viability [9]. If a bank cannot meet its financial obligations, it could threaten its stability and negatively impact the financial system as a whole. Banks distribute deposit funds into loan funds to financial markets, such as individ- uals or companies [27] to finance various projects, investments, and economic activities [12]. The amount of loans given to individuals or companies also depends on the amount of deposits in the bank. Banks often have loan limits as a form of credit risk control [32]. Consequently, the more companies or individuals that request a loan, the higher competition rate, which may slow down the process or reduce their chances of successfully obtaining a loan. Before giving a loan, the bank will conduct a credit analysis to ensure that the potential credit recipient is truly trustworthy so that it reduces the possibility of bad credit or Non-Performing Loan (NPL). Credit analysis includes the background of potential credit recipients, business prospects, and guarantees that include various indem- nification agreements if the loan is not returned on time [1, 34]. Loans that are not repaid on time or are not repaid at all are called to as NPL [10]. If NPL occurs, it will affect the availability of funds in the bank [21] as well as its profitability and bank operations [7, 13]. The distribution of deposits into loans is analogous to the predator-prey interaction model. The predator-prey interaction model was first introduced by the American bio- physicist Lotka in 1925 and by the Italian mathematician Volterra in 1926 [19]. In a predator-prey interaction, the predator will eat the prey. This is similar to the behaviour of the distribution of deposit and loan funds where loans as predators and deposits as prey. In 2013, Haque et al. proposed a two predators-one prey model by adding the effects of intra-specific competition that occurs in each predator [11]. Furthermore, Mukhopadhyay and Bhattacharyya (2016) constructed a two predators-one prey model by adding the ef- fect of inter-predator competition to the model [18]. Mukhopadhyay and Bhattacharyya also added the effect of harvesting on one of the predators. Harvesting occurs due to interactions with humans, when humans take or kill predator or prey for consumption or sale that have been used primarily in fishing, forestry, and wildlife conservation. In [8, 23, 24, 30], the effect of harvesting on predator-prey interactions was studied. Deposit withdrawals in banks exhibit behaviour similar to prey harvesting. Furthermore, Long et al. (2022) developed a two predators-one prey model using the effects of inter-specific and intra-specific competition [16]. This is also in accordance with the previous problem, where there is competition between loan volumes in the bank. In 2014, Sumarti et al. proposed a dynamic model of deposit and loan volumes based on the Lotka-Volterra predator-prey model [29]. One of the deposit and loan volumes models of Sumarti et al. (2014) uses the Michaelis-Menten response function. Then, in 2018, Sumarti et al. proposed a bank balance sheet model [28]. After that, in 2021, Ansori O. F. Damayanti, I. Darti, A. Suryanto / Eur. J. Pure Appl. Math, 18 (1) (2025), 5643 3 of 24 et al. modified the parameters used in the deposit and loan fund variables of the Sumarti et al. (2018) balance sheet model [4]. In deposit volume, Ansori et al. (2021) considered the rate of deposit withdrawal, while for loan volume, the NPL rate and the loan repayment rate were added. The increase in deposit volume is assumed to follow logistic growth because of the bank’s limitations in receiving funds from customers [3, 4, 29]. The deposit and loan model of Sumarti et al. (2014) has been developed into a model of two loans, individual and company at one bank that compete each other for getting loan volumes, similar to the two predators-one prey model of Long et al. (2022). Then, the deposit volume will be considered the effect of withdrawal by adding a withdrawal rate based on the research of Ansori et al. (2021). In addition, for both loan volumes, the effect of NPL on both loan volumes will also be considered by adding the NPL rate and the loan repayment rate based on the research of Ansori et al. (2021). Thus, this research proposes a model of two loans in one bank that considers the effects of NPL and deposit withdrawal. We set the rest of the paper as follows: In Section 2, we develope the model, followed by the verification of the existence, uniqueness, non-negativity, and boundness solutions of the model in Section 3. The existence of equilibrium point and its local stability are discussed in Sections 4 and 5, while the global stability analysis of the equilibrium points is given in Section 6. Moreover, in Section 7, we demonstrate some numerical simulations with a set of hypothetical parameters to validate the theoretical results. Finally, the conclusion is presented in Section 8. 2. The Mathematical Model The model in this paper is constructed by developing deposit and loan volumes model of Sumarti et al. (2014) into a model of two loans at one bank which consists of three variables, namely D which represents the volume of deposit, L1 which represents the volume of individual loan, and L2 which represents the volume of company loan. Then, in the deposit volume, a parameter of withdrawal rate (w) is added, while in the both loan volumes, parameters of NPL (ni) and repayment rate of overdue loans (ci(1 − µi)) are added according to the research of Ansori et al. (2021). The increase of deposit volume uses a logistic growth model because banks have limitations in managing deposit funds effectively. The distribution of deposit volume into loan volumes uses the Michaelis-Menten response function due to the limited deposit volumes that can be lent to customers. In this model, the deposit interest rate is the only factor that affects the increase of deposit and the depositor will not be able to withdraw more than their deposit volume and additional interest, so that the deposit interest rate (γ) are always greater than the rate of withdrawal (w). The nonlinear system autonomous model of two loans at one bank is obtained as follows. dD dt = γD ( 1− D k ) − p1DL1 b1D + 1 − p2DL2 b2D + 1 − wD, O. F. Damayanti, I. Darti, A. Suryanto / Eur. J. Pure Appl. Math, 18 (1) (2025), 5643 4 of 24 dL1 dt = p1DL1 b1D + 1 − (σ1 + n1)L1 − c1(1− µ1)L1 − β1L1L2 − h1L 2 1, (1) dL2 dt = p2DL2 b2D + 1 − (σ2 + n2)L2 − c2(1− µ2)L2 − β2L1L2 − h2L 2 2, with non-negative initial condition D(0), L1(0), and L2(0). The description of the param- eters in model (1) can be seen in Table 1 where i = 1 for individual loan and i = 2 for company loan. The proposed model resembles the two predators-one prey model of Long et al. (2022), and some analyses will follow their approach. Table 1: Description of Parameters Parameter Description bi The barriers to obtaining loans ci Loan repayments rate h1 Competition rate between individual loans h2 Competition rate between company loans k The carrying capacity of deposits ni Non-performing loan rate pi Maximum rate of the mixture between deposit and loan volumes w Withdrawal deposit rate β1 Competition rate of two loans on individual loan β2 Competition rate of two loans on company loan γ Interest rate of deposit µi The average portion of NPL σ1 Interest rate of loan 3. Preliminaries Results As model (1) represents the volume of funds in the bank, the solution of the model must exist and unique, non-negative, and ultimately bounded. The following theorems ensure that the solution of the model (1) satisfies them. Theorem 1. All solutions of system (1) exist and are unique in Ω× [0,∞) for any non- negative initial condition D(0), L1(0), and L2(0) where Ω = {(D,L1, L2) ∈ R3 + ∪ {⃗0} : max{|D|, |L1|, |L2|} ≤ M} for sufficiently large of M . Proof: Let the existence of M be guaranteed by the boundedness of the solution which will be proved in the next theorem. It is also assumed that D(t), L1(t), L2(t) ≥ 0 for every t ≥ 0 which will also be proved in the next theorem. Let X = (D,L1, L2), X̄ = (D̄, L̄1, L̄2) and consider a mapping F (X) = (F1(X), F2(X), F3(X)) such that F1(X) = γD ( 1− D k ) − p1DL1 b1D + 1 − p2DL2 b2D + 1 − wD, O. F. Damayanti, I. Darti, A. Suryanto / Eur. J. Pure Appl. Math, 18 (1) (2025), 5643 5 of 24 F2(X) = p1DL1 b1D + 1 − (σ1 + n1)L1 − c1(1− µ1)L1 − β1L1L2 − h1L 2 1, F3(X) = p2DL2 b2D + 1 − (σ2 + n2)L2 − c2(1− µ2)L2 − β2L1L2 − h2L 2 2. Then, for any X, X̄ ∈ Ω, we have ||F (X)− F (X̄)|| = |F1(X)− F1(X̄)|+ |F2(X)− F2(X̄)|+ |F3(X)− F3(X̄)| ≤ ( γ + 2γM k + w + 2p1M + 2p2M ) |D − D̄| +(2p1M +H5 + β1M + 2h1M + β2M) |L1 − L̄1| +(2p2M +H6 + β1M + 2h2M + β2M) |L2 − L̄2|, = K1|D − D̄|+K2|L1 − L̄1|+K3|L2 − L̄2| ≤ K ( |D − D̄|+ |L1 − L̄1|+ |L2 − L̄2| ) = K||X − X̄||, where K1 = γ + 2γM k + w + 2p1M + 2p2M, K2 = H5 +M(2p1 + β1 + 2h1 + β2), K3 = H6 +M(2p2 + β1 + 2h2 + β2), K = max{K1,K2,K3}, H5 = σ1 + n1 + c1(1− µ1), and H6 = σ2 + n2 + c2(1− µ2). Based on a Lemma in [22] (p.71), the function F (X) satisfies the Lipschitz condition so that the system (1) has a unique solution in Ω× [0,∞). Theorem 2. All solutions of system (1) are non-negative and ultimately bounded for any non-negative initial condition D(0), L1(0), and L2(0). Proof: If we assume that D(t) can be negative. This shows that there exists t1 > 0 such that D(t) > 0 for 0 < t < t1, D(t) = 0 for t = t1 and D(t) < 0 for t > t1. Based on the system (1) obtained dD dt ∣∣∣∣∣ t=t1 = 0, meaning that there is no rate of increase in the deposit fund when t = t1. This contradicts the statement that D(t) < 0 for t > t1, so the supposition is incorrect, which implies that D(t) ≥ 0 for every t > 0. In the same way, L1(t) ≥ 0 and L2(t) ≥ 0 for any t > 0. Furthermore, suppose V (t) = D(t) + L1(t) + L2(t), then the first derivative of V (t) is as follows. dV dt = (γ − w)D − γD2 k + (c1µ1 − c1 − σ1 − n1)L1 + (c2µ2 − c2 − σ2 − n2)L2 O. F. Damayanti, I. Darti, A. Suryanto / Eur. J. Pure Appl. Math, 18 (1) (2025), 5643 6 of 24 −(β1 + β2)L1L2 − h1L 2 1 − h2L 2 2 ≤ (γ − w)D − γD2 k + (c1µ1 − c1 − σ1 − n1)L1 + (c2µ2 − c2 − σ2 − n2)L2. Then, for every positive constant ξ is obtained dV dt + ξV ≤ (γ + ξ − w)D − γD2 k + (ξ + c1µ1 − c1 − σ1 − n1)L1 +(ξ + c2µ2 − c2 − σ2 − n2)L2. by taking ξ < min{c1(1− µ1) + σ1 + n1, c2(1− µ2) + σ2 + n2}, then we get dV dt + ξV ≤ (γ + ξ − w)D − γD2 k ≤ k(γ + ξ − w)2 4γ , dV dt ≤ k(γ + ξ − w)2 4γ − ξV. Clearly, the solution to a first order differential inequality satisfies V (t) ≤ k(γ + ξ − w)2 4ξγ + ( V (0)− k(γ + ξ − w)2 4ξγ ) e−ξt. It is obvious that V (t) → k(γ + ξ − w)2 4ξγ as t → ∞. Thus, all solutions of system (1) are ultimately bounded to the region Ω = { {D(t), L1(t), L2(t)} ∈ R3 + ∪ {⃗0} : V (t) ≤ { k(γ + ξ − w)2 4ξγ }} . 4. The Existence of The Equilibrium Points The equilibrium points of the system (1) are obtained when dD dt = dL1 dt = dL2 dt = 0. (2) By solving the system (2), five equilibrium points are obtained, namely (i) The equilibrium point without deposit and loan activities, E0 = (0, 0, 0), that always exists. (ii) The loan-free equilibrium point, E1 = ( k γ (γ − w), 0, 0 ) , that always exists since γ > w. (iii) The individual loan-free equilibrium point, E2 = (D(2), 0, L2(2)). L2(2) exist ifH4(2) > H6 where O. F. Damayanti, I. Darti, A. Suryanto / Eur. J. Pure Appl. Math, 18 (1) (2025), 5643 7 of 24 L2(2) = 1 h2 ( p2D(2) 1 + b2D(2) − (σ2 + n2 + c2(1− µ2)) ) and H4(2) = p2D(2) 1 + b2D(2) . L2(2) still contains D(2) while D(2) values are the roots of the following cubic equation D3 (2) + 3B1D 2 (2) + 3B2D(2) +B3 = 0, (3) where B1 = −h2b2[(γ − w)kb2 − 2γ] 3γh2b22 , B2 = −h2[2(γ − w)kb2 − γ] + kp2(p2 −H6b2) 3γh2b22 , B3 = −k[(γ − w)h2 + p2H6] γh2b22 . Let χ = D(2) +B1, then (3) can be written as χ3 + 3ϖ1χ+ϖ2 = 0, (4) where ϖ1 = B2 −B2 1 , ϖ2 = B3 − 3B1B2 + 2B3 1 . Based on Cardan criterion in [6], the possible positive roots of equation (4) are as follows. (a) If ϖ2 < 0, then eq. (4) has a single positive root. (b) Suppose ϖ2 > 0, ϖ1 < 0 and i. if ϖ2 2 + 4ϖ3 1 = 0, then eq. (4) has a positive root of multiplicity two, ii. if ϖ2 2 + 4ϖ3 1 < 0, then eq. (4) has two positive roots. (c) If ϖ2 = 0 and ϖ1 < 0, then Eq. (4) has a unique positive root. Thus, the third equilibrium point E2 = (D(2), 0, L2(2)) exists if H4(2) > H6 and it satisfies one of the Cardan criterion. (iv) The company loan-free equilibrium point, E3 = (D(3), L1(3) , 0). The existence of this equilibrium point also uses the Cardan criterion [6] in the same way as E2. So, E3 exists if H3(3) > H5 and it satisfies one of the Cardan criterion, where L1(3) = 1 h1 ( p1D(3) 1 + b1D(3) − (σ1 + n1 + c1(1− µ1)) ) and H3(3) = p1D(3) 1 + b1D(3) . (v) The equilibrium point with deposit and loan activities, E∗ = (D∗, L∗ 1, L ∗ 2), where L∗ 1 = p1h2(D ∗ + b2D ∗2)− β1p2(D ∗ + b1D ∗2) + (1 + b1D ∗)(1 + b2D ∗)[β1H6 −H5h2] (1 + b1D∗)(1 + b2D∗)(h1h2 − β1β2) O. F. Damayanti, I. Darti, A. Suryanto / Eur. J. Pure Appl. Math, 18 (1) (2025), 5643 8 of 24 and L∗ 2 = p2h1(D ∗ + b1D ∗2)− β2p1(D ∗ + b2D ∗2) + (1 + b1D ∗)(1 + b2D ∗)[β2H5 −H6h1] (1 + b1D∗)(1 + b2D∗)(h1h2 − β1β2) . L∗ 1 and L∗ 2 exist if they satisfy one of these conditions. (a) If h1h2 > β1β2, then it must satisfy h2 β1 > H∗ 4 −H6 H∗ 3 −H5 > β2 h1 . (5) (b) If h1h2 < β1β2, then it must satisfy h2 β1 < H∗ 4 −H6 H∗ 3 −H5 < β2 h1 . (6) Further, D∗ values are the roots of the following equation G0D ∗5 +G1D ∗4 +G2D ∗3 +G3D ∗2 +G4D ∗ +G5 = 0 (7) where G0 = −γ(h1h2 − β1β2)b 2 1b 2 2, G1 = (h1h2 − β1β2)b1b2[k(γ − w)b1b2 − 2γ(b1 + b2)], G2 = (h1h2 − β1β2)[2kb1b2(γ − w)(b1 + b2)− γ(b21 + b22 + 4b1b2)] +kp1b2[b1b2(h2H5 − β1H6) + b1p2(β1 + β2)− p1b2h2] +kp2b1[b1b2(h1H6 − β2H5)− p2b1h1], G3 = (h1h2 − β1β2)[k(γ − w)(b21 + b22 + 4b1b2)− 2γ(b1 + b2)] +kp1[(2b1b2 + b22)(h2H5 − β1H6) + p2(β1 + β2)(b1 + b2)− 2p1b2h2] +kp2[(2b1b2 + b21)(h1H6 − β2H5)− 2p2b1h1], G4 = (h1h2 − β1β2)[2k(γ − w)(b1 + b2)− γ] +kp1[(b1 + 2b2)(h2H5 − β1H6) + p2(β1 + β2)− p1h2] +kp2[(2b1 + b2)(h1H6 − β2H5)− p2h1], G5 = k(γ − w)(h1h2 − β1β2) + k[p1(h2H5 − β1H6) + p2(h1H6 − β2H5)]. By Descartes’ Rule of Signs in [33], equation (7) will have at least one positive real root. Thus, E∗ exists if it satisfies (5) or (6). 5. Local Stability The Jacobian matrix at the equilibrium point is obtained as follows. J(D,L1, L2) =  J1 −H3 −H4 H1 J2 −β1L1 H2 −β2L2 J3  , (8) O. F. Damayanti, I. Darti, A. Suryanto / Eur. J. Pure Appl. Math, 18 (1) (2025), 5643 9 of 24 where J1 = γ ( 1− 2D k ) −H1 −H2 − w, J2 = H3 −H5 − β1L2 − 2h1L1, J3 = H4 −H6 − β2L1 − 2h2L2, H1 = p1L1 (b1D + 1)2 , H2 = p2L2 (b2D + 1)2 , H3 = p1D b1D + 1 , H4 = p2D b2D + 1 , H5 = σ1 + n1 + c1(1− µ1), and H6 = σ2 + n2 + c2(1− µ2). The next theorems give the local stability of the equilibrium points by evaluating the real part of all eigenvalues of the Jacobian matrix (8). Theorem 3. The equilibrium point without deposit and loan activities E0 of the system (1) is always unstable. Proof: The Jacobian matrix at E0 is given by J(0, 0, 0) =  γ − w 0 0 0 −σ1 − n1 − c1(1− µ1) 0 0 0 −σ2 − n2 − c2(1− µ2)  . It is clear that the eigenvalues are λ1 = γ − w, λ2 = −σ1 − n1 − c1(1− µ1) < 0, λ3 = −σ2 − n2 − c2(1− µ2) < 0. Since γ > w, then λ1 > 0. Hence, the equilibrium point without deposit and loan activities E0 is always unstable. Theorem 4. The loan-free equilibrium point E1 of the system (1) is locally asymptotically stable if p1D(1) b1D(1) + 1 < H5 and p2D(1) b2D(1) + 1 < H6. Proof: The Jacobian matrix at E1 is given by J(D(1), 0, 0) =  −(γ − w) − p1D(1) b1D(1) + 1 − p2D(1) b2D(1) + 1 0 p1D(1) b1D(1) + 1 −H5 0 0 0 p2D(1) b2D(1) + 1 −H6  . It is clear that the eigenvalues are λ1 = −(γ − w) < 0, λ2 = p1D(1) b1D(1) + 1 − H5, and λ3 = p2D(1) b2D(1) + 1 −H6. If p1D(1) b1D(1) + 1 < H5 and p2D(1) b2D(1) + 1 < H6, then λ2 < 0 and λ3 > 0 O. F. Damayanti, I. Darti, A. Suryanto / Eur. J. Pure Appl. Math, 18 (1) (2025), 5643 10 of 24 consequently E1 is locally asymptotically stable. Otherwise, it is an unstable point. So, if the maximum loanable funds are too low relative to the three main factors reducing loan funds, lending activity will cease, potentially harming the bank and preventing further loans. Theorem 5. By letting the individual loan-free equilibrium point E2 of the system (1) exists. If H3(2) −H5 < β1L2(2) and γ k > H2(2)b2, then E2 is locally asymptotically stable, where H2(2) and H3(2) are stated in the proof. Proof: The Jacobian matrix at E2 is given by J(D(2), 0, L2(2)) =  D(2) ( H2(2)b2 − γ k ) −H3(2) −H4(2) 0 H3(2) −H5 − β1L2(2) 0 H2(2) −β2L2(2) −h2L2(2)  , where H2(2) = p2L2(2) (b2D(2) + 1)2 , H3(2) = p1D(2) b1D(2) + 1 , and H4(2) = p2D(2) b2D(2) + 1 . The characteristic equation of the Jacobian matrix J(E2) is given by( H3(2) −H5 − β1L2(2) − λ ) ∣∣J(2) − λI ∣∣ = 0 where J(2) = ( D(2) ( H2(2)b2 − γ k ) −H4(2) H2(2) −h2L2(2) ) . It is clear that the first eigenvalue is λ1 = H3(2) − H5 − β1L2(2) which will negative if H3(2) − H5 < β1L2(2) . Then, the other two eigenvalues λ2 and λ3 will be negative if tr(J(2)) < 0 and |J(2)| > 0. If γ k > H2(2)b2, then tr(J(2)) < 0 and |J(2)| > 0. Thus, if H3(2) −H5 < β1L2(2) and γ k > H2(2)b2, then E2 is locally asymptotically stable. So, if the maximum loanable funds for individuals are too low compared to factors reducing them, individual lending will cease, leaving the bank without profit from such loans. Theorem 6. Assume that the company loan-free equilibrium point E3 of the system (1) exists. If H4(3) −H6 < β2L1(3) and γ k > H1(3)b1, then E3 is locally asymptotically stable, where H4(3) and H1(3) are stated in the proof. Proof: The Jacobian matrix at E3 is given by J(D(3), L1(3) , 0) =  D(3) ( H1(3)b1 − γ k ) −H3(3) −H4(3) H1(3) −h1L1(3) −β1L1(3) 0 0 H4(3) −H6 − β2L1(3)  , where O. F. Damayanti, I. Darti, A. Suryanto / Eur. J. Pure Appl. Math, 18 (1) (2025), 5643 11 of 24 H1(3) = p1L1(3) (b1D(3) + 1)2 , H3(3) = p1D(3) b1D(3) + 1 , and H4(3) = p2D(3) b2D(3) + 1 . And we have the characteristic equation of the Jacobian matrix J(E3) as follows.( H4(3) −H6 − β2L1(3) − λ ) ∣∣J(3) − λI ∣∣ = 0, where J(3) = ( D(3) ( H1(3)b1 − γ k ) −H3(3) H1(3) −h1L1(3) ) . It is obvious that the first eigenvalue is λ1 = H4(3) −H6−β2L1(3) . If H4(3) −H6 < β2L1(3) , then λ1 < 0. Next, the other two eigenvalues will be negative if tr(J(3)) < 0 and |J(3)| > 0. If γ k > H1(3)b1, then tr(J(3)) < 0 and |J(3)| > 0. Hence, if H4(3) − H6 < β2L1(3) and γ k > H1(3)b1, then E3 is locally asymptotically stable. Thus, if the maximum amount of funds available for the company loan is too low relative to the factors that reduce lending, the company lending will stop, preventing the bank from making profit from this loan. Theorem 7. If E∗ exists, a1 > 0, a3 > 0 and a1a2−a3 > 0 where a1, a2 and a3 are stated in the proof, then E∗ is locally asymptotically stable. Proof: The Jacobian matrix at E∗ is given by J(D∗, L∗, L∗) =  D∗ ( H∗ 1b1 +H∗ 2b2 − γ k ) −H∗ 3 −H∗ 4 H∗ 1 −h1L ∗ 1 −β1L ∗ 1 H∗ 2 −β2L ∗ 2 −h2L ∗ 2  , where H∗ 1 = p1L ∗ 1 (b1D∗ + 1)2 , H∗ 2 = p2L ∗ 2 (b2D∗ + 1)2 , H∗ 3 = p1D ∗ b1D∗ + 1 , andH∗ 4 = p2D ∗ b2D∗ + 1 By evaluating |J(E∗)−λI| = 0, we obtain the characteristic equation of the matrix J(E∗) as follows. λ3 + a1λ 2 + a2λ+ a3 = 0, (9) where a1 = j2 + j5 − j1, a2 = j2j5 +H∗ 2H ∗ 4 +H∗ 1H ∗ 3 − j1j2 − j1j5 − j3j4, a3 = j1j3j4 + j2H ∗ 2H ∗ 4 + j5H ∗ 1H ∗ 3 − j1j2j5 − j3H ∗ 2H ∗ 3 − j4H ∗ 1H ∗ 4 . j1 = D∗ ( H∗ 1b1 +H∗ 2b2 − γ k ) , j2 = h1L ∗ 1, j3 = β1L ∗ 1, j4 = β2L ∗ 2, j5 = h2L ∗ 2. Based on the Routh-Hurwitz criterion in [19], the solution of equation (9) has a negative real part if and only if a1 > 0, a3 > 0 and a1a2 − a3 > 0. Therefore, E∗ is locally asymptotically stable if a1 > 0, a3 > 0 and a1a2 − a3 > 0. O. F. Damayanti, I. Darti, A. Suryanto / Eur. J. Pure Appl. Math, 18 (1) (2025), 5643 12 of 24 6. Global Stability 6.1. Global Stability of E1 Theorem 8. E1 is globally asymptotically stable (GAS) if p1D(2) ≤ H5 and p2D(1) ≤ H6. Proof: We first consider a Lyapunov function. U = D −D(1) −D(1) ln D D(1) + L1 + L2 The first derivative of U with respect to t is given by dU dt = (D −D(1)) ( γD(1) k − γD k − p1L1 b1D + 1 − p2L2 b2D + 1 ) + ( p1DL1 b1D + 1 −H5L1 − β1L1L2 − h1L 2 1 ) + ( p2DL2 b2D + 1 −H6L2 − β2L1L2 − h2L 2 2 ) = −γ k (D −D(1)) 2 + ( p1D(1) b1D + 1 −H5 ) L1 + ( p2D(1) b2D + 1 −H6 ) L2 −(β1 + β2)L1L2 − h1L 2 1 − h2L 2 2 ≤ −γ k (D −D(1)) 2 + ( p1D(1) −H5 ) L1 + ( p2D(1) −H6 ) L2 −(β1 + β2)L1L2 − h1L 2 1 − h2L 2 2. If p1D(1) ≤ H5 and p2D(1) ≤ H6, then dU dt ≤ 0 for all (D,L1, L2) ∈ R3 + ∪ {⃗0} and dU dt = 0 for (D,L1, L2) = (D(1), 0, 0). According to the LaSalle invariance principle in [2], E1 is GAS. 6.2. Global Stability of E2 Theorem 9. E2 is GAS if γ k > p2b2L2(2) and p1D(2) + v1β2L2(2) ≤ H5. Proof: By considering a Lyapunov function as follows. V1 = D −D(2) −D(2) ln D D(2) + L1 + v1 ( L2 − L2(2) − L2(2) ln L2 L2(2) ) , where v1 = b2D(2) + 1. Then, the first derivative of V1 is given by dV1 dt = (D −D(2)) ( γD(2) k + p2L2(2) b2D(2) + 1 + w − γD k − p1L1 b1D + 1 − p2L2 b2D + 1 − w ) + ( p1DL1 b1D + 1 −H5L1 − β1L1L2 − h1L 2 1 ) +v1(L2 − L2(2)) ( p2D b2D + 1 − p2D(2) b2D(2) + 1 + h2L2(2) − β2L1 − h2L2 ) , O. F. Damayanti, I. Darti, A. Suryanto / Eur. J. Pure Appl. Math, 18 (1) (2025), 5643 13 of 24 dV1 dt = −γ k (D −D(2)) 2 − v1h2(L2 − L2(2)) 2 + p2b2L2(2)(D −D(2)) 2 (b2D(2) + 1)(b2D + 1) + p2(D −D(2))(L2 − L2(2))[v1 − (b2D(2) + 1)] (b2D(2) + 1)(b2D + 1) − (v1β2 + β1)L1L2 + ( p1D(2) b1D + 1 −H5 + v1β2L2(2) ) L1 − h1L 2 1 ≤ −γ k (D −D(2)) 2 − v1h2(L2 − L2(2)) 2 + p2b2L2(2)(D −D(2)) 2 + p2(D −D(2))(L2 − L2(2))[v1 − (b2D(2) + 1)] (b2D(2) + 1)(b2D + 1) − (v1β2 + β1)L1L2 + ( p1D(2) −H5 + v1β2L2(2) ) L1 − h1L 2 1, since v1 = b2D(2) + 1, it can be written as dV1 dt ≤ −( γ k − p2b2L2(2))(D −D(2)) 2 − v1h2(L2 − L2(2)) 2 − (v1β2 + β1)L1L2 + ( p1D(2) −H5 + v1β2L2(2) ) L1 − h1L 2 1. If γ k > p2b2L2(2) and p1D(2)+ v1β2L2(2) ≤ H5, then dV1 dt ≤ 0 for all (D,L1, L2) ∈ R3 + ∪ {⃗0} and dV1 dt = 0 for (D,L1, L2) = (D(2), 0, L2(2)). Applying the LaSalle invariance principle in [2], E2 is GAS. 6.3. Global Stability of E3 Theorem 10. E3 is GAS if γ k > p1b1L1(3) and p2D(3) + v2β1L1(3) ≤ H6. Proof: We consider a Lyapunov function as follows. V2 = D −D(3) −D(3) ln D D(3) + v2 ( L1 − L1(3) − L1(3) ln L1 L1(3) ) + L2, where v2 = b1D(3) + 1 The first order derivative of the Lyapunov function V2 is given by dV2 dt = (D −D(3)) ( γ ( 1− D k ) − p1L1 b1D + 1 − p2L2 b2D + 1 − w ) +v2(L1 − L1(3)) ( p1D b1D + 1 − (σ1 + n1 + c1(1− µ1))− β1L2 − h1L1 ) + ( p2DL2 b2D + 1 − (σ2 + n2 + c2(1− µ2))L2 − β2L1L2 − h2L 2 2 ) , O. F. Damayanti, I. Darti, A. Suryanto / Eur. J. Pure Appl. Math, 18 (1) (2025), 5643 14 of 24 dV2 dt = −γ k (D −D(3)) 2 − v2h1(L1 − L1(3)) 2 + p1b1L1(3)(D −D(3)) 2 (b1D(3) + 1)(b1D + 1) + p1(D −D(3))(L1 − L1(3))[v2 − (b1D(3) + 1)] (b1D(3) + 1)(b1D + 1) − (v2β1 + β2)L1L2 + ( p2D(3) b2D + 1 −H6 + v2β1L1(3) ) L2 − h2L 2 2 ≤ −γ k (D −D(3)) 2 − v2h1(L1 − L1(3)) 2 + p1b1L1(3)(D −D(3)) 2 + p1(D −D(3))(L1 − L1(3))[v2 − (b1D(3) + 1)] (b1D(3) + 1)(b1D + 1) − (v2β1 + β2)L1L2 + ( p2D(3) −H6 + v2β1L1(3) ) L2 − h2L 2 2, = −( γ k − p1b1L1(3))(D −D(3)) 2 − v2h1(L1 − L1(3)) 2 − (v2β1 + β2)L1L2 + ( p2D(3) −H6 + v2β1L1(3) ) L2 − h2L 2 2, If γ k > p1b1L1(3) and p2D(3) + v2β1L1(3) ≤ H6, then dV2 dt ≤ 0 for all (D,L1, L2) ∈ R3 + ∪ {⃗0} and dV2 dt = 0 for (D,L1, L2) = (D(3), L1(3) , 0). According to the LaSalle invariance principle in [2], E3 is GAS. 6.4. Global Stability of E∗ Theorem 11. E∗ is GAS if k > p1b1L ∗ 1 + p2b2L ∗ 2 and 0 < β1 2h2−β2 < v4 v3 < 2h1−β1 β2 where v3 and v4 are stated in the proof. Proof: Define a Lyapunov function as L = D −D∗ −D∗ ln D D∗ + v3 ( L1 − L∗ 1 − L∗ 1 ln L1 L∗ 1 ) + v4 ( L2 − L∗ 2 − L∗ 2 ln L2 L∗ 2 ) , where v3 = b1D ∗ + 1 and v4 = b2D ∗ + 1. The first order derivative of L is dL dt = (D −D∗) ( − γ k (D −D∗)− p1(L1(b1D ∗ + 1)− L∗ 1(b1D + 1)) (b1D + 1)(b1D∗ + 1) −p2(L2(b2D ∗ + 1)− L∗ 2(b2D + 1)) (b2D + 1)(b2D∗ + 1) ) +v3(L1 − L∗ 1) ( p1(D −D∗) (b1D + 1)(b1D∗ + 1) − β1(L2 − L∗ 2)− h1(L1 − L∗ 1) ) +v4(L2 − L∗ 2) ( p2(D −D∗) (b2D + 1)(b2D∗ + 1) − β2(L1 − L∗ 1)− h2(L2 − L∗ 2) ) , O. F. Damayanti, I. Darti, A. Suryanto / Eur. J. Pure Appl. Math, 18 (1) (2025), 5643 15 of 24 dL dt = −γ k (D −D∗)2 + (D −D∗) ( p1[b1L ∗ 1(D −D∗)− (L1 − L∗ 1)(b1D ∗ + 1)] (b1D + 1)(b1D∗ + 1) ) +(D −D∗) ( p2[b2L ∗ 2(D −D∗)− (L2 − L∗ 2)(b2D ∗ + 1)] (b2D + 1)(b2D∗ + 1) ) + v3p1(D −D∗)(L1 − L∗ 1) (b1D + 1)(b1D∗ + 1) + v4p2(D −D∗)(L2 − L∗ 2) (b2D + 1)(b2D∗ + 1) −v3h1(L1 − L∗ 1) 2 − v4h2(L2 − L∗ 2) 2 − (v3β1 + v4β2)(L1 − L∗ 1)(L2 − L∗ 2) = −γ k (D −D∗)2 + p1b1L ∗ 1(D −D∗)2 (b1D + 1)(b1D∗ + 1) + p2b2L ∗ 2(D −D∗)2 (b2D + 1)(b2D∗ + 1) + p1(D −D∗)(L1 − L∗ 1)[v3 − (b1D ∗ + 1)] (b1D + 1)(b1D∗ + 1) + p2(D −D∗)(L2 − L∗ 2)[v4 − (b2D ∗ + 1)] (b2D + 1)(b2D∗ + 1) −v3h1(L1 − L∗ 1) 2 − v4h2(L2 − L∗ 2) 2 − (v3β1 + v4β2)(L1 − L∗ 1)(L2 − L∗ 2) ≤ −γ k (D −D∗)2 + p1b1L ∗ 1(D −D∗)2 + p2b2L ∗ 2(D −D∗)2 + p1(D −D∗)(L1 − L∗ 1)[v3 − (b1D ∗ + 1)] (b1D + 1)(b1D∗ + 1) + p2(D −D∗)(L2 − L∗ 2)[v4 − (b2D ∗ + 1)] (b2D + 1)(b2D∗ + 1) −v3h1(L1 − L∗ 1) 2 − v4h2(L2 − L∗ 2) 2 − (v3β1 + v4β2)(L1 − L∗ 1)(L2 − L∗ 2), = − (γ k − p1b1L ∗ 1 − p2b2L ∗ 2 ) (D −D∗)2 − v3h1(L1 − L∗ 1) 2 − v4h2(L2 − L∗ 2) 2 −(v3β1 + v4β2)(L1 − L∗ 1)(L2 − L∗ 2). Let P = √ v3β1 + v4β2 2 (L1 − L∗ 1) and Q = √ v3β1 + v4β2 2 (L2 − L∗ 2), then we have P 2 +Q2 − (P +Q)2 = −2PQ = −(v3β1 + v4β2)(L1 − L∗ 1)(L2 − L∗ 2), so that dL dt ≤ − (γ k − p1b1L ∗ 1 − p2b2L ∗ 2 ) (D −D∗)2 − v3h1(L1 − L∗ 1) 2 − v4h2(L2 − L∗ 2) 2 + v3β1 + v4β2 2 (L1 − L∗ 1) 2 + v3β1 + v4β2 2 (L2 − L∗ 2) 2 − (√ v3β1 + v4β2 2 (L1 − L∗ 1) + √ v3β1 + v4β2 2 (L2 − L∗ 2) )2 ≤ − (γ k − p1b1L ∗ 1 − p2b2L ∗ 2)(D −D∗)2 − v3h1(L1 − L∗ 1 )2 − v4h2(L2 − L∗ 2) 2 + v3β1 + v4β2 2 (L1 − L∗ 1) 2 + v3β1 + v4β2 2 (L2 − L∗ 2) 2, O. F. Damayanti, I. Darti, A. Suryanto / Eur. J. Pure Appl. Math, 18 (1) (2025), 5643 16 of 24 dL dt = − (γ k − p1b1L ∗ 1 − p2b2L ∗ 2 ) (D −D∗)2 − ( v3h1 − v3β1 + v4β2 2 ) (L1 − L∗ 1) 2 − ( v4h2 − v3β1 + v4β2 2 ) (L2 − L∗ 2) 2. If γ k > p1b1L ∗ 1 + p2b2L ∗ 2 and 0 < β1 2h2−β2 < v4 v3 < 2h1−β1 β2 , then dL dt ≤ 0 for all (D,L1, L2) ∈ R3 + ∪ {⃗0} and dL dt = 0 for (D,L1, L2) = (D∗, L∗ 1, L ∗ 2). By applying the Lasalle invariance principle in [2], E∗ is GAS. 7. Numerical Simulation Table 2: Parameters Values Parameter Simulation 1 Simulation 2 Simulation 3 Simulation 4 γ 0.4 0.4 0.4 0.4 w 0.2 0.2 0.2 0.2 k 3 3 3 3 p1 0.25 0.35 0.8 0.85 p2 0.35 0.8 0.35 0.9 n1 0.17 0.17 0.17 0.17 n2 0.23 0.23 0.23 0.2 b1 0.8 0.8 0.8 0.9 b2 0.9 0.7 0.9 0.95 σ1 0.1 0.2 0.1 0.2 σ2 0.15 0.1 0.15 0.15 c1 0.2 0.2 0.2 0.15 c2 0.25 0.2 0.25 0.2 β1 0.5 0.5 0.5 0.1 β2 0.4 0.4 0.4 0.15 h1 0.1 0.1 0.1 0.3 h2 0.1 0.1 0.1 0.25 µ1 0.15 0.15 0.15 0.15 µ2 0.2 0.1 0.2 0.2 In this section, we perform some numerical simulation results of the model (1) using the fourth-order Runge-Kutta scheme in [15], to confirm our theoretical results. In [17, 20], the fourth-order Runge-Kutta method is used to solve the first-order ordinary differential equation corresponding to the model in this paper. This method is also widely used for the numerical simulation of dynamical system models such as in [25, 31]. All parameter values are chosen hypothetically because of the unavailability of the real data work. O. F. Damayanti, I. Darti, A. Suryanto / Eur. J. Pure Appl. Math, 18 (1) (2025), 5643 17 of 24 The parameter values of model (1) are shown in Table 2 with the initial condition D(0) = 10, L1(0) = 4, L2(0) = 5 are used for simulation 1,2,3, while L1(0) = 6, L2(0) = 3 are used for simulation 4. The first simulation is conducted using the values of the parameters in the simulation 1 column of the Table 2. After the calculation, we find that the solution of the system is asymptotically stable to the loan-free equilibrium point, E1 = (1.5, 0, 0). The numerical simulation shown in Figure 1, which also shows that the solution converges to E1. Figure 1: Numerical Simulation 1 Figure 2: Numerical Simulation 2 O. F. Damayanti, I. Darti, A. Suryanto / Eur. J. Pure Appl. Math, 18 (1) (2025), 5643 18 of 24 Figure 3: Numerical Simulation 3 Figure 4: Numerical Simulation 4 The second simulation is conducted using the parameters values in the simulation 2 column of the Table 2. We find that the system solution is asymptotically stable to the individual loan-free equilibrium point E2 = (1.1904, 0, 0.0946), which also shown in the numerical simulation in Figure 2. In simulation 3, we see that the solution of the system will be asymptotically stable to the company loan-free equilibrium point, E3 = (1.0396, 0.1405, 0) which also shown in the numerical simulation in Figure 3. Lastly, the simulation uses the parameter values in the simulation 4 column of the Table 2. Using these parameters, we find that the solution of the system is asymptotically stable to the equilibrium point with deposit and loan activities, E∗ = (1.3021, 0.0273, 0.0391). The O. F. Damayanti, I. Darti, A. Suryanto / Eur. J. Pure Appl. Math, 18 (1) (2025), 5643 19 of 24 numerical simulation shown in Figure 4 also shows that the solution converges to E∗. The simulation shows that if the maximum loanable funds for individual and company exceed all factors reducing them, lending activity will continue. In reality, the loan-free equilibrium point, E1, is an equilibrium point that is highly avoided by a bank because in the absence of loans, a banks will not have the profit of interest loan that used for operational costs and return interest on depositors. By observing the parameter changes in the previous numerical simulations, it can be seen that if the maximum rate of mixture between deposit and both loan volumes or loanable volume is very high, there will eventually be individual loan and company loan. The amount of loanable volume depends on the amount of deposit volume. If the deposit volume can be properly distributed into the loan, then the banking operation will be stable so that the equilibrium point with the deposit and loan activities, E∗, is the most desirable condition for a bank. Figure 5: The Effect of Changes in Withdrawal Rate on The Volume of (a) Deposit (b) Individual Loan (c) Company Loan O. F. Damayanti, I. Darti, A. Suryanto / Eur. J. Pure Appl. Math, 18 (1) (2025), 5643 20 of 24 Figure 6: The Effect of Changes in Individual’s NPL Rate on The Volume of (a) Deposit (b) Individual Loan (c) Company Loan The effect of changes in withdrawal and NPL rates on deposit and both loan volumes is presented in Figures 5, 6, and 7. Figure 5 shows the effect of withdrawal on the deposit volume. If the rate of withdrawal is getting bigger, then the volume of deposit volume will decrease. It can lead to a reduced volume of loan that can be lent to individuals and companies. Moreover, Figure 6 shows the effect of individual’s NPL on deposit and both loan volumes. The higher individual’s NPL rate, the greater volumes of deposit and company loan, while the volume of individual loan decreases. This is in accordance with what happens in the bank, if someone has a high rate of NPL, then that person will get less loan volume which causes the volume of loan can be lent to the companies and there is more deposit volume left. Furthermore, Figure 7 shows the effect of the company’s NPL rate on the deposit volume and both loan volumes. The higher rate of company’s NPL, the higher deposit and individual loan volumes, but the company loan volume is less. This is also in accordance with what actually happens in the bank, if a company has a high NPL rate, it will receive fewer loans, which allows more deposit volume to be lent to individuals. O. F. Damayanti, I. Darti, A. Suryanto / Eur. J. Pure Appl. Math, 18 (1) (2025), 5643 21 of 24 Figure 7: The Effect of Changes in Company’s NPL Rate on The Volume of (a) Deposit (b) Individual Loan (c) Company Loan The results of this study cannot be directly implemented in banking policies. However, they can serve as a consideration for banks in formulating policies or operational strate- gies. For instance, banks may adjust their strategies to increase deposits through product promotion or to reduce the volume of loans. These actions could indirectly influence the stability of the banking system. 8. Conclusion In this paper, we construct a model of two loans in one bank analogous to the two predators-one prey model and consider withdrawal and NPL rates. The existence and uniqueness solution of the model (1) have been proved. Moreover, the non-negativity and boundedness of the solution have also been proved. The model (1) has five equilibrium points, namely the equilibrium point without deposit and loan activities, the loan-free equilibrium point, the individual loan-free equilibrium point, the company loan-free equi- librium point, and the equilibrium point with deposit and loan activities. The equilibrium O. F. Damayanti, I. Darti, A. Suryanto / Eur. J. Pure Appl. Math, 18 (1) (2025), 5643 22 of 24 point without deposit and loan activities always exists and is unstable. The loan-free equi- librium point always exists and is globally asymptotically stable if it satisfies two certain conditions. The individual loan-free equilibrium point and the company loan-free equilib- rium point are asymptotically stable, both locally and globally, if they satisfy their certain conditions. The equilibrium point with deposit and loan activities is globally asymptot- ically stable if some certain conditions are satisfied. Our theoretical results have been confirmed by numerical solutions of the model. The results of the model presented in this paper can be implemented if all assumptions are met. Therefore, for future research, we recommend that the parameters used are derived from real-world data. Acknowledgements This work was supported by Faculty of Mathematics and Natural Science (FMIPA) through Public Funds DPA (Dokumen Pelaksanaan Anggaran) Perguruan Tinggi Berbadan Hukum Pelaksanaan Anggaran) Perguruan Tinggi Berbadan Hukum (PTNBH) University of Brawijaya and based on FMIPA Professor Grant, with contract number: 02172.22/UN10. F0901/B/KS/2024, dated July 12, 2024. References [1] T Abdullah and S Wahjusaputri. Bank dan Lembaga Keuangan. Mitra Wacana Media, Jakarta, Indonesia, 2018. [2] K T Alligood, T D Sauer, and J A Yorke. CHAOS: An Introduction to Dynamical Systems. Springer-Verlag, New York, USA, 2000. [3] M F Ansori, K A Sidarto, and N Sumarti. Logistic Models of Deposit and Loan between Two Banks with Saving and Debt Transfer Factors. AIP Publishing, 2192(1), 2019. [4] M F Ansori, K A Sidarto, N Sumarti, and I Gunadi. Dynamics of Bank’s Balance Sheet: A System of Deterministic and Stochastic Differential Equation Approach. International Journal of Mathematics and Computer Science, 16(3):871–884, 2021. [5] D R Bhattarai. Capital Adequacy Ratio and Financial Performance of Commercial Banks in Nepal. Tribhuvan University Journal, 36(1):96–105, 2021. [6] Y Cai, C Zhao, W Wang, and J Wang. Dynamics of A Leslie–Gower Predator–Prey Model with Additive Allee Effect. Applied Mathematical Modelling, 39(7):2092–2106, 2015. [7] G N Chukwu, T A Muritala, J O Akande, and A O Adekunle. Impact of Non- Performing Loan on Bank Performance in Nigeria. Journal of Law and Sustainable Development, 12(6), 2024. [8] A Das and S K Roy. Dynamics of Stage-Structured Prey–Predator Model with Prey Refuge and Harvesting. International Journal of Modelling and Simulation, 42(6):966–984, 2022. [9] F D Dzapasi. The impact of Liquidity Management on Bank Financial Performance O. F. Damayanti, I. Darti, A. Suryanto / Eur. J. Pure Appl. Math, 18 (1) (2025), 5643 23 of 24 in a Subdued Economic Environment: A Case of the Zimbabwean Banking Industry. PM World Journal, 9(1):1–20, 2020. [10] H V Greuning and S B Bratanovic. Analyzing Banking Risk: A Framework for Assessing Corporate Governance and Risk Management. World Bank Publications, Washington DC, USA, 2020. [11] M I Haque, N Ali, and S Chakravarty. Study of A Tri-Trophic Prey-Dependent Food Chain Model of Interacting Populations. Mathematical Biosciences, 246(1):55–71, 2013. [12] N I Hasan. Pengantar Perbankan. Gaung Persada Press Group, Jakarta, Indonesia, 2018. [13] S O Khalifaturofi’ah. Cost Efficiency, Innovation and Financial Performance of Banks in Indonesia. Journal of Economic and Administrative Sciences, 39(1):100–116, 2023. [14] M Kumhof and X Wang. Banks, Money, and The Zero Lower Bound on Deposit Rates. Journal of Economic Dynamics and Control, 132, 2021. [15] S Lenhart and J T Workman. Optimal Control Applied to Biological Models. CRC Press, Taylor Francis Group, London, UK, 2007. [16] Y Long, LWang, and J Li. Uniform Persistence and Multistability in A Two Predator- One Prey System with Inter-specific and Intra-specific Competition. Journal of Ap- plied Mathematics and Computing, pages 1–28, 2022. [17] S T Muhammad. Pengkajian Metode Extended Runge Kutta dan Penerapannya pada Persamaan Diferensial Biasa. Jurnal sains dan seni ITS, 4(2), 2016. [18] B Mukhopadhyay and R Bhattacharyya. Effects of Harvesting and Predator Inter- fence in A Model of Two-Predators Competing for A Single Prey. Applied Mathemat- ical Modelling, 40(4):3264–3274, 2016. [19] J D Murray. Mathematical Biologi: An Introduction. Springer-Verlag, New York, USA, 2002. [20] W Pandia and I Sitepu. Penentuan Galat Persamaan Diferensial Biasa Orde 1 dengan Metode Numerik. Jurnal Mutiara Pendidikan Indonesia, 6(1):31–37, 2021. [21] C Y Park and K Shin. COVID-19, Nonperforming Loans, and Cross-border Bank Lending. Journal of Banking Finance, 133, 2021. [22] L Perko. Differential Equations and Dynamical Systems. Springer Science Business Media, New York, USA, 2001. [23] A S Purnomo, I Darti, and A Suryanto. Dynamics of Eco-Epidemiological Model with Harvesting. AIP Conference Proceedings, 1913(1), 2017. [24] K Pusawidjayanti, A Suryanto, and R B E Wibowo. Dynamics of A Predator-Prey Model Incorporating Prey Refuge, Predator Infection and Harvesting. Applied Math- ematical Sciences, 9(76):3751–3760, 2015. [25] H F Romli, A Suryanto, and I Darti. Dynamics of Foot-and-Mouth Disease Spread Model for Cattle with Carrier, Vaccination, and Environmental Transmission. Com- munnications in Mathematical Biology and Neuroscience, 2023, 2023. [26] A Ruslan. Capital, Bank size, Credit Risk and Bank Performance. International Journal of Innovative Science and Research Technology, 4(5), 2019. [27] N Sumarti and M F Ansori. Model Matematika Perbankan. ITB Press, Bandung, O. F. Damayanti, I. Darti, A. Suryanto / Eur. J. Pure Appl. Math, 18 (1) (2025), 5643 24 of 24 Indonesia, 2022. [28] N Sumarti, A Fadhlurrahman, and H R Widyani. The Dynamical System of The Deposit and Loan Volumes of A Commercial Bank Containing Interbank Lending and Saving Factors. Southeast Asian Bullet of Mathematics, 42(5), 2018. [29] N Sumarti, R Nurfitriyana, and W Nurweda. A Dynamical System of Deposit and Loan Volumes Based on The Lotka-Volterra Model. AIP Conference Proceedings, 1587(1):92–94, 2014. [30] A Suryanto and I Darti. Stability Analysis and Nonstandard Grunwald-Letnikov Scheme for A Fractional Order Predator-Prey Model with Ratio-Dependent Func- tional Response. International Journal of Mathematics and Mathematical Sciences, 2019(1), 2019. [31] A Suryanto, I Darti, and Trisilowati. Dynamic Behavior of A Harvested Logistic Model Incorporating Feedback Control. Communnications in Mathematical Biology and Neuroscience, 2024, 2024. [32] W Wang and G Cai. Curtailing Bank Loan and Loan Insurance Under Risk Regula- tions in Supply Chain Finance. Management Science, 70(4):2682–2698, 2024. [33] S Wiggins. Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer-Verlag, New York, USA, 2003. [34] T M Yhip and B M Alagheband. Credit Analysis and Credit Management. The Practice of Lending: A Guide to Credit Analysis and Credit Risk, pages 3–46, 2020.