mathematis Artile Prey Predator Models with Variable Carrying Capaity Mariam K. A. Al-Moqbali, Nasser S. Al-Salti and Ibrahim M. Elmojtaba * Department of Mathematis and Statistis, College of Siene, Sultan Qaboos University, Musat 123, Oman; m010852@student.squ.edu.om M.K.A.A.-M.); nalsalti@squ.edu.om N.S.A.-S.) * Correspondene: elmojtaba@squ.edu.om Reeived: 8 April 2018; Aepted: 13 June 2018; Published: 15 June 2018 Abstrat: Prey predator models with variable arrying apaity are proposed. These models are more realisti in modeling population dynamis in an environment that undergoes hanges. In partiular, prey predator models with Holling type I and type II funtional responses, inorporating the idea of a variable arrying apaity, are onsidered. The arrying apaity is modeled by a logisti equation that inreases sigmoidally between an initial value κ 0 > κ 1 a lower bound for the arrying apaity) and a final value κ 1 + an upper bound for the arrying apaity). In order to examine the effet of the variable arrying apaity on the prey predator dynamis, the two models were analyzed qualitatively using stability analysis and numerial solutions for the prey, and the predator population densities were obtained. Results on global stability and Hopf bifuration of ertain equilibrium points have been also presented. Additionally, the effet of other model parameters on the prey predator dynamis has been examined. In partiular, results on the effet of the handling parameter and the predator s death rate, whih has been taken to be the bifuration parameter, are presented. Keywords: prey predator; variable arrying apaity; limit yle; hopf bifuration 1. Introdution Prey predator dynami is an essential tool in mathematial eology, speifially for our understanding of interating populations in the natural environment. This relationship will ontinue to be one of the dominant themes in both eology and mathematial eology due to its universal existene and importane. These problems may appear to be mathematially simple at first sight. However, they are, in fat, often very hallenging and ompliated. Moreover, although muh progress has been made to the prey predator theory in the last 40 years, many long standing mathematial and eologial problems remain open, suh as modeling transient dynamis, environmental variability, omplex eologial networks, and biodiversity extrapolation tehniques [1]. All populations are affeted by hanges in their environment; therefore, there is a need to treat the arrying apaity as a system variable i.e., funtion of time) in order to model population dynamis in an environment that undergoes hanges [2]. In partiular, in resoure management, where the arrying apaity is often assumed to be onstant and unhanging [3]. Many efforts to predit the world s arrying apaity, the maximum sustainable population, are based on this assumption [4]. However, tehnologial developments have raised rop yields, allowing a greater population to be supported by a smaller land area [5]. Thus, for the human population, a onstant arrying apaity is not realisti [6]. Similarly, in nature, the inherent variability of natural systems [7] means that assuming an unhanging arrying apaity fails to adequately represent the environment. Meyer et al. [8] proposed the arrying apaity to be modeled by a logisti equation that inreases sigmoidally between an initial value κ 0 > κ 1 and a final value κ 1 +. They studied the effet of this dynami arrying apaity on the trajetories of simple growth models, and they use the new model to Mathematis 2018, 6, 102; doi:10.3390/math6060102 www.mdpi.om/journal/mathematis
Mathematis 2018, 6, 102 2 of 12 re-analyze two atual ases of the growth of human populations; English and Japanese examples with two pulses, or one hange in limit, appear to verify their model. A periodi form of arrying apaity has also been onsidered. For example, Shepherd et al. [9] has onsidered the following expression for the arrying apaity κεt) = κ 0 + δsinεt) where κ 0 and δ are positive onstants suh that δ < κ 0, ensuring the positive of κt), and ε is small and positive. They demonstrated that, when the arrying apaity varies slowly with time, a multiple time sale analysis leads to approximate losed form solutions that, apart from being expliit, are omparable to numerially generated ones and are valid for a range of parameter values. In this paper, we will onsider prey predator models with Holling type I and type II funtional responses, inorporating the idea of a variable arrying apaity. Note that Meyer et al. [8] and Shepherd et al. [9] only onsider the variable arrying apaity within a single population, and here we onsider prey predator models with variable arrying apaity. The rest of the paper is organized as follows: in the seond setion, we present and analyze a prey predator model with Holling type I funtional response and variable arrying apaity. In Setion 3, we present and analyze a prey predator model with Holling type II funtional response and variable arrying apaity. Finally, a onlusion is given in Setion 4. 2. Prey Predator Model with Holling Type I Funtional Response If the predator eats essentially one type of prey, then the funtional response should be linear at low prey density. Hene, in this setion, we will take the Lotka Voltera model with the onept of variable arrying apaity. 2.1. Model Building The prey predator model with Holling type I funtional response with logisti arrying apaity is governed by the following system of equations: dnt) = rnt) 1 Nt) ) ant)pt) κt) dκt) = α κt) κ 1 ) 1 κt) κ ) 1 dpt) = bnt)pt) Pt) subjet to the initial onditions: N0) = N 0, κ0) = κ 0, P0) = P 0, where N, P denote prey and predator population densities, respetively, r represents prey s per apita growth rate, is the death rate of the predator, b represents the inrement of predator and a represents derements of prey and κ is the arrying apaity that inreases sigmoidally between an initial value κ 0 > κ 1 and a final value κ 1 + with a growth rate α. 2.2. Mathematial Analysis of the Model System 1) has the following equilibrium points: E 1 = 0, κ 1, 0), E 2 = κ 1,κ 1, 0), E 3 = 0, κ 1 +, 0), E 4 = κ 1 +, κ 1 +, 0), E 5 = b, κ 1, rbκ ) 1 ), and E 6 = abκ 1 b, κ 1 +, rbκ ) 1 + ) ). abκ 1 + ) Note that E 5 and E 6 exist iff bκ 1 > 0 and bκ 1 + ) > 0, respetively. Obviously, the arrying apaity, κt), is independent of the other two variables and always tends to κ 1 + ; therefore, the instability of the equilibrium points E 1, E 2, and E 5 immediately follows. The loal stability of the remaining equilibrium points is illustrated in the following theorem. 1)
Mathematis 2018, 6, 102 3 of 12 Theorem 1. The stability of the equilibrium points E 3, E 4, and E 6 of System 1) is given by i) E 3 = 0, κ 1 +, 0) is unstable. ii) E 4 = κ 1 +, κ 1 +, 0) is loally asymptotially stable if bκ 1 + ) <. iii) E 6 = b, κ 1 +, rbκ ) 1 + ) ) is loally asymptotially stable if bκ abκ 1 + ) 1 + ) >. Proof. The Jaobian matrix of System 1) is Therefore, J = r 2rN κ ap rn 2 an 0 α 2ακ + 2ακ 1 0 bp 0 bn. i) the eigenvalues of the Jaobian at E 3 are r, α, and, whih implies that E 3 is also unstable; ii) the eigenvalues of the Jaobian at E 4 are r, α, and bκ 1 + ), so E 4 is stable if bκ 1 + ) < ; iii) the Jaobian matrix at E 6 is given by J ) b,κ 1 +, rbκ 1 +b ) = abκ 1 + ) r bκ 1 + ) r 2 b 2 κ 1 + ) 2 a b 0 α 0 rbκ 1 + b ) aκ 1 + ) 0 0 Clearly, α is one of the eigenvalues, so the remaining two eigenvalues are the eigenvalues of the redued matrix: r a bκ 1 + ) b whih has the harateristi polynomial: λ 2 + rbκ 1 + b ) aκ 1 + ) 0, r bκ 1 + ) λ + rbκ 1 + b ) = 0. aκ 1 + ) Using Routh Hurwitz Criteria [10], the loal stability is guaranteed if bκ 1 + ) >.. The following theorem shows the global stability of the equilibrium point E 4 in the N κ plane. Theorem 2. The equilibrium point E 4 is globally asymptotially stable in the positive quadrant of N κ plane if bκ 1 + ) <. Proof. In the N κ plane, the system is redued to
Mathematis 2018, 6, 102 4 of 12 dnt) = rnt) 1 Nt) ) = g κt) 1 N, κ) dκt) = α κt) κ 1 ) 1 κt) κ ) 1 = g 2 N, κ). 2) 1 Let D = Nκ κ 1 ). Clearly, D > 0 in the interior of the positive quadrant of N κ plane, as κ > κ 1 in the interior. Now let N, κ) = N Dg 1) + κ Dg 2) r = κκ κ 1 ) α. N Therefore, does not hange sign and it is not identially zero in the positive quadrant of N κ plane; therefore, from Dula s riteria [11], there exists no limit yle in the positive quadrant of N κ plane. From the loal stability of E 4, we onlude the proof. Note that, although E 4 is globally stable in the N κ phase plane of the system 2), it an still be unstable in the 3D phase spae of the full System 1) 2.3. Numerial Simulation and Disussion Numerial simulations of System 1) are illustrated in Figure 1 for i) bκ 1 + ) < and ii) bκ 1 + ) >. Figure 1i) illustrates the ase when the prey and the predator reah the stable equilibrium point E 4 = κ 1 +, κ 1 +, 0); that is, the prey follows the urve of arrying apaity, whereas the predator is extint if > bκ 1 + ). Figure 1ii) demonstrates the ase where the prey and the predator reah the stable equilibrium point E 6 = b, κ 1 +, rbκ ) 1 + ) ) with < bκ abκ 1 + ) 1 + ). The prey and predator populations exhibit damped osillations before reahing the asymptotially stable spiral equilibrium point. These dynamis are similar to the onstant arrying apaity ase. However, if the growth rate of variable arrying apaity is very small, then the periodiity in the solutions of the systems derease its magnitude. Additionally, the solutions reah the stable equilibrium faster ompared to the onstant arrying apaity ase and ompared to the variable arrying apaity ase with a higher growth rate, as illustrated in Figure 2. In addition, Figure 3 illustrates the effet of variable arrying apaity on the phase spae of the model, whih onfirms that less time is needed for the prey and predator populations to reah their equilibrium values when the growth rate of arrying apaity is very small, whih an be seen from the trajetories in the neighbourhood of the asymptotially stable spiral point. i) a = b = 0.00025, > bκ 1 + ) ii) a = b = 0.0025, < bκ 1 + ) Figure 1. Prey predator model with Holling type I funtional response and logisti arrying apaity, stability of E 4 and E 6 for κ 1 = 300, = 500, r = = 0.25, α = 0.2.
Mathematis 2018, 6, 102 5 of 12 i) Constant apaity, κ = 800 ii) κt), α = 0.4 iii) κt), α = 0.008 Figure 2. Effet of the logisti variable apaity on the prey predator model with linear response for κ 1 = 300, = 500, r = = 0.25, a = b = 0.0025. i) Constant apaity, κ = 800 ii) κt), α = 0.4 iii) κt), α = 0.008 Figure 3. Effet of the logisti variable varying apaity on the phase spae of the prey and predator model with linear response for κ 1 = 300, = 500, r = = 0.25, a = b = 0.0025. 3. Prey Predator Model with Holling Type II Funtional Response A predator has to devote a ertain time to searh, ath, and onsume its prey. If the prey density inreases then searhing beomes easier, but onsuming prey takes the same amount of time. The funtional response is therefore an inreasing funtion of the prey density, obviously equal to zero at zero prey density, approahing a finite time at high densities. If the predator hunts different types of prey, then the funtional response should inrease as a power greater than 1 usually 2). Aording to funtional response, it an be expressed that there should exist a saturation effet; that is, the predator s birth rate should tend toward a finite limit at high prey densities. Here, in this setion, we onsider the prey predator model with Holling type II funtional response with variable arrying apaity. 3.1. Model Building The prey predator model with Holling type II funtional response with logisti arrying apaity is governed by the following system of equations:
Mathematis 2018, 6, 102 6 of 12 dnt) = rnt) 1 Nt) ) ant)pt) κt) 1 + γnt) dκt) = α κt) κ 1 ) 1 κt) κ ) 1 dpt) = bnt)pt) 1 + γnt) Pt) 3) where γ is the handling parameter. 3.2. Mathematial Analysis of the Model System 3) has six equilibrium points; namely: E 1 = ) 0, κ 1, 0), E 2 = κ 1,κ 1, 0), E 3 = 0, κ 1 +, 0), ) E 4 = κ 1 +, κ 1 +, 0), E 5 = b γ), κ 1, rb[κ 1b γ) )], E aκ 1 b γ) 2 6 = b γ), κ 1 +, rb[κ 1+ )b γ) )]. aκ 1 + )b γ) 2 Note that E 5 and E 6 exist iff b γ) > 0 and κ 1 + )b γ) >, respetively. As noted before, the arrying apaity is independent of the other two variables, so the equilibrium points E 1, E 2, and E 5 are unstable; the loal stability of the remaining equilibrium points is illustrated in the following theorem. Theorem 3. The loal stability of the equilibrium points E 3, E 4, and E 6 of System 3) is given by the following. i) E 3 = 0, κ 1 +, 0) is unstable. ii) E 4 = κ 1 +, κ 1 +, 0) is loally asymptotially ) stable if κ 1 + )b γ) <. iii) E 6 = b γ), κ 1 +, rb[κ 1+ )b γ) )] is loally asymptotially stable if γκ aκ 1 + )b γ) 2 1 + )b γ) b γ < 0. Proof. The Jaobian matrix of the system is given by J = r 2rN κ ap 1 Nγ ) 1 + γn 1 + γn rn 2 ) κ2 + 2κ 0 α 1 2κ bp 1 γn ) 1 + γn 1 + γn 0 an 1 + γn 0 bn 1 + γn By evaluating the Jaobian matrix at eah equilibrium point, we have the following. i) The eigenvalues of the Jaobian matrix of System 3) at E 3 are r, α, and, so E 3 is also unstable. ii) The eigenvalues of the Jaobian matrix of System 3) at E 4 are r, α, and κ 1 + )b γ). Clearly, if κ 1 + )b γ) < 0, then E 4 is stable. iii) The Jaobian matrix of the system at the equilibrium point E 6 is given by J ) = b γ),κ 1+, rbb γ)κ 1 + ) ) ab γ) 2 κ 1 + ) rγκ 1 + )b γ) b γ) bκ 1 + )b γ) r 2 b γ) 2 κ 1 + ) 2 a b 0 α 0 rb γ)κ 1 + ) ) aκ 1 + ) 0 0 Clearly, α is one of the eigenvalues. The other two eigenvalues are the eigenvalues of the redued matrix: J = r γκ 1 + )b γ) b γ) bκ 1 + )b γ) r b γ)κ 1 + ) ) aκ 1 + ) a b 0...
Mathematis 2018, 6, 102 7 of 12 where its harateristi polynomial is λ 2 r γκ 1 + )b γ) b γ) λ + r κ 1 + )b γ) ) = 0 bκ 1 + )b γ) bκ 1 + ) Using Routh Hurwitz Criteria, in order for this point to be loally stable, we should have: κ 1 + )b γ) > 0 and γκ 1 + )b γ) b γ < 0. The global stability for the equilibrium point E 4 is stated in the following theorem. Theorem 4. The equilibrium point E 4 = κ 1 +, κ 1 +, 0) is globally stable if b γ)κ 1 + ) < in the positive quadrant of the N κ plane. Proof. The proof is similar to the proof of Theorem 2. Note that, although E 4 is globally stable in the 2D phase plane, it an still be unstable in the 3D phase spae of the full System 3) The following theorem states the possibility of ourrene of Hopf bifuration. Theorem 5. System 3) undergoes a Hopf bifuration at the positive equilibrium E 6 when = 0 = b γκ 1 + ) 1) γ κ 1 + ) + 1). Proof. The eigenvalues of the linearized system around the equilibrium point E 6 are α and µ 1,2 = α) ± iβ) where α) = 1 2 traj ) β) = detj ) α)) 2. and Now, at 0, α 0 ) = 0, β 0 ) = rb γκ 1 + ) 1) γ κ 1 + ) + 1) κ 1 + )γ 2 = 0 dα d = 0 = r 1 + γκ 1 + )) γκ 1 + ) 1) = 0. 4bκ 1 + ) Therefore, from Hopf Theorem [12], the proof is onluded. 3.3. Numerial Simulation and Disussion Numerial simulations of System 3) are illustrated in Figure 4 for i) bκ 1 + )b γ) < and ii) γκ 1 + )b γ) b γ < 0. Figure 4i) illustrates the ase where the prey and the predator reah the stable equilibrium point E 4 = κ 1 +, κ 1 +, 0); that is, the prey follows the urve of arrying apaity, whereas the predator is extint if > b γ)κ 1 ) + ). While Figure 4ii) demonstrates the stability of E 6 = b γ), κ 1 +, rb[κ 1+ )b γ) )] if < b γ)κ aκ 1 + )b γ) 2 1 + ) with b γ > 0, b + γ and γ <. We an see from Figure 4 that the predator and prey populations exhibit κ 1 + )b γ) damped osillations. Moreover, these osillations derease in magnitude for smaller values of the arrying apaity growth rate as illustrated in Figure 5. This effet an also be seen in Figure 6, whih demonstrates the phase spae of the model where we have a stable limit yle. However, when we redue the growth rate to be muh smaller, the effet of the variable arrying apaity is to onvert the stable limit yle into a spirally stable equilibrium point as shown in Figure 7. In addition, when we redue the handling parameter γ, the prey and the predator populations exhibit damped osillations before reahing an asymptotially stable spiral equilibrium point and the effet of variable
Mathematis 2018, 6, 102 8 of 12 arrying apaity is the same as in the Holling type I, that is, the periodiity in the solutions of the systems derease its magnitude and reah to the stable equilibrium faster than the systems with onstant arrying apaity as shown in Figures 8 and 9. The effet of the death rate of the predator, whih is the bifuration parameter, on the prey and predator population dynamis is illustrated in Figures 10 and 11. It an be seen that, when the death rate is small up to around = 0.3) the populations are periodi and reah a stable limit yle. If the death rate is higher up to around = 0.7), the populations exhibit damped osillations and reah the asymptotially stable spiral equilibrium point E 6, whih represents the o-existene of both populations. If the death rate is even higher than around 0.75, then the two populations reah the asymptotially stable point E 4, whih represents the extintion of the predator and the existene of only the prey. i) a = b = 0.00025 ii) a = b = 0.0025 Figure 4. Prey predator model with Holling type II funtional response and logisti arrying apaity, stability of E 4 and E 6, κ 1 = 300, = 500, r = = 0.25, γ = 0.002, α = 0.2. i) Constant apaity, κ = 800 ii) κt), α = 0.4 iii) κt), α = 0.008 Figure 5. Effet of logisti arrying apaity on prey predator model with Holling type II funtional response for κ 1 = 300, = 500, r = = 0.25, a = b = 0.0025, γ = 0.002.
Mathematis 2018, 6, 102 9 of 12 i) Constant apaity, κ = 800 ii) κt), α = 0.4 iii) κt), α = 0.008 Figure 6. Effet of the logisti variable arrying apaity on the phase spae of the prey predator model with Holling type II funtional response for κ 1 = 300, = 500, r = = 0.25, a = b = 0.0025, γ = 0.002. i) κt), α = 0.002 ii) κt), α = 0.002 Figure 7. Effet of logisti arrying apaity on prey predator model with Holling type II funtional response, κ 1 = 300, = 500, r = = 0.25, a = b = 0.0025, γ = 0.002. i) Constant apaity, κ = 800 ii) κt), α = 0.4 iii) κt), α = 0.008 Figure 8. Effet of logisti arrying apaity on prey predator model with Holling type II funtional response, κ 1 = 300, = 500, r = = 0.25, a = b = 0.0025, γ = 0.0002.
Mathematis 2018, 6, 102 10 of 12 i) Constant apaity, κ = 800 ii) κt), α = 0.4 iii) κt), α = 0.008 Figure 9. Effet of the logisti variable arrying apaity on the phase spae of the prey and predator model with Holling type II funtional response for κ 1 = 300, = 500, r = = 0.25, a = b = 0.0025, γ = 0.0002. i) κt), = 0.28 ii) κt), = 0.38 iii) κt), = 0.77 Figure 10. Effet of the bifuration parameter on the prey predator model with Holling type II funtional response, κ 1 = 300, = 500, r = 0.3, a = b = 0.0025, γ = 0.002, m = 0.2, α = 0.02. i) κt), = 0.28 ii) κt), = 0.38 iii) κt), = 0.77 Figure 11. Effet of the bifuration parameter on the phase spae of the prey predator model with Holling type II funtional response with κ 1 = 300, = 500, r = 0.3, a = b = 0.0025, γ = 0.002, m = 0.2, α = 0.02. 4. Conlusions Our aim in this paper is to examine the effet of variable arrying apaity on the prey predator dynamis. For this purpose, two prey predator models with logisti arrying apaity have been onsidered, namely, prey predator models with Holling type I and type II funtional responses.
Mathematis 2018, 6, 102 11 of 12 Stability analysis and numerial solutions of the two models were obtained, and the results were displayed graphially. Our results show that, when interation between the prey and the predator populations was onsidered to be of Holling type I funtional response, the prey and predator populations exhibit damped osillations before reahing the asymptotially stable spiral equilibrium point. However, if the growth rate of the variable arrying apaity is very small, then the osillations derease in magnitude. The solutions were also found to reah the stable equilibrium faster ompared to the onstant arrying apaity ase and ompared to the variable arrying apaity ase with higher growth rate. In addition, the equilibrium point representing existene of prey and the extintion of the predator with maximum arrying apaity was found to be globally asymptotially stable under a ertain ondition in the N κ plane; however, it might not be stable in the 3D plane. In the ase when the interation between the prey and the predator was onsidered to follow Holling type II funtional response, our results showed that, when a limit yle exists and when the growth rate of the logisti variable arrying apaity is very small, the stable limit yle onverted into a spirally stable equilibrium point. Our results also showed that this system undergoes a Hopf bifuration at the positive equilibrium representing the o-existene of prey and predator with the maximum arrying apaity under a ertain value of death rate. The equilibrium point representing the existene of prey and the extintion of predator with the maximum arrying apaity was found to be globally asymptotially stable under a ertain ondition in the N κ plane. It is worth nothing that this might not be the ase in the 3D plane. Moreover, when the handling parameter was redued, the prey and the predator were found to exhibit damped osillations and then reahed an asymptotially stable spiral equilibrium point. The effet of variable arrying apaity in this ase is the same as in in Holling type I; that is, the periodiity in the solutions of the system dereases in magnitude and reahes the stable equilibrium faster than the system with onstant arrying apaity. Finally, when the death rate of predators, whih was taken to be the bifuration parameter, was inreased, the prey and the predator dynamis hanged from having periodi behaviour that, by exhibiting damped osillations and reahing an asymptotially stable spiral equilibrium point representing the o-existene of both populations, reahed a stable limit yle for small values of the death rate of the predator to a situation where the two populations reah an asymptotially stable point representing the existene of the prey population only in the ase of the higher predator s death rate. Author Contributions: I.M.E. and N.S.A.-S. proposed and formulated the models; M.K.A.A.-M. arried out the mathematial analysis and the numerial simulations; M.K.A.A.-M., N.S.A.-S., and I.M.E. wrote the paper. Aknowledgments: M.K.A.A.-M. aknowledges with thanks the support from the Ministry of Eduation, Sultanate of Oman. The authors would like to aknowledge the support form Sultan Qaboos University. The authors would also like to thank two anonymous reviewers whose omments improved the quality and readability of the paper. Conflits of Interest: The authors delare that there is no onflit of interest. Referenes 1. Green, J.L.; Hastings, A.; Arzberger, P.; Ayala, F.J.; Cottingham, K.L.; Cuddington, K.; Davis, F.; Dunne, J.A.; Fortin, M.; Gerber, L.; et al. Complexity in Eology and Conservation: Mathematial, Statistial, and Computational Challenges. BioSiene 2005, 55, 501 510. [CrossRef] 2. Krebs, C.J. Eology the Experimental Analysis of Distribution and Abundane; Pearson Eduation, In.: Upper Saddle River, NJ, USA, 2009. 3. Monte-Luna, D.; Brook, B.W.; Zetina-Rejón, M.J.; Cruz-Esalona, V.H. The arrying apaity of eosystems. Glob. Eol. Biogeogr. 2004, 13, 485 495. [CrossRef] 4. Cohen, J.E. How many people an the earth support? Sienes 1995, 35, 18 23. [CrossRef] 5. Waggoner, P.E. How muh land an ten billion people spare for nature? Daedalus 1995, 125, 73 93. 6. Meershaert, M.M. Mathematial Modelling; Aademi Press: Waltham, MA, USA, 2013.
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