A PREY-PREDATOR MODEL WITH HOLLING II FUNCTIONAL RESPONSE AND THE CARRYING CAPACITY OF PREDATOR DEPENDING ON ITS PREY

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1 Journal of Applied Analsis and Computation Volume 8, Number 5, October 218, Website: DOI: / A PREY-PREDATOR MODEL WITH HOLLING II FUNCTIONAL RESPONSE AND THE CARRYING CAPACITY OF PREDATOR DEPENDING ON ITS PREY Hanwu Liu 1,, Ting Li 1,2 and Fengqin Zhang 1 Abstract One pre-predator model is formulated and the global behavior of its solution is analzed. In this model, the carring capacit of predator depends on the amount of its pre, and the Holling II functional response is involved. This model ma have four classes of positive equilibriums and limit ccle. The positive equilibriums ma be stable, or a saddle-node, or a saddle, or a degenerate singular point. In alpine meadow ecosstem, the dnamics of vegetation and plateau pika can be described b this model. Through simulating with virtual parameters, the cause of alpine meadow degradation and effective recover strateg is investigated. Increasing grazing rate or decreasing plateau pika mortalit ma cause alpine meadow degradation. Correspondingl, reducing grazing rate and increasing plateau pika mortalit ma recover the degraded alpine meadow effectivel. ewords Pre-predator model, densit-dependent, global behavior, Holling II functional response. MSC(21) 34D2, 34C6, 92D Introduction In ecosstem, predation is an important relationship among populations. A lot of predator-pre models have been investigated [14, 16]. Taking into account the digestive saturation of predator, Holling proposed three functional response functions in 1965 [6], and the Holling II functional response is incorporated in man models [4, 7, 8, 17] thereafter. In general, the growth of predator is assumed to be densit-dependent [4, 8, 18, 19], and the carring capacit is assumed to be constant [1, 9, 11, 15]. However, due to the complicated relationships in ecosstem, the carring capacit of predator ma be influenced b the amount of its pre. For eample, in alpine meadow ecosstem, plateau pika feeds on vegetation, at the same time, vegetation is also the main component of plateau pika living environment. the corresponding author. address: liuhanwu-china@163.com(h. Liu) 1 Department of Applied Mathematics, Yuncheng Universit, Yuncheng 44, China 2 School of Mathematics and Computer Science, Shani Normal Universit, Linfen 414, China The authors were supported b National Natural Science Foundation of China ( , ), Research Project Supported b Shani Scholarship Council of China ( ) and Project of Yuncheng Universit (sws2164).

2 A Pre-Predator model with Holling II functional response Because the higher vegetation prevents plateau pika from finding its natural enem, so environment with higher vegetation is not suitable for plateau pika to live in. Liu [1] indicates that the carring capacit of plateau pika increases firstl and then decreases with vegetation heightening. There are few research considering model in which the carring capacit of predator depends on the amount of its pre. Falcone etc [5] proposed the model (1.1) = r α, (1.1) = d + β f() 2, where, (t) and (t) are the number of pre and predator at time t respectivel. Pre population grows eponentiall with growth rate r >. In the absence of pre, the mortalit rate of predator is d >. Parameter α > is predation rate of predator and β > is the conversion rate. The growth of predator is densitdependent, and its carring capacit is 1 f() which depends on the amount of its pre. However, there was not theoretical analsis in [5], Falcone and Israel [4] analzed model (1.1) preliminaril, Chu and Ding [2] and Tedeschini-Lalli [13] studied model (1.1) numericall to show the intertwined attractive basins of two sinks. In second section, we formulate a pre-predator model and analze its global behavior. In this model, the carring capacit of predator depends on the amount of its pre, and the Holling tpe II functional response is involved. The third section provides four eamples and corresponding numerical simulation. In section 4, we analze the reason of alpine meadow degradation and eplore the effective recover strategies. Section 5 is a discussion. 2. Model formulation and analsis When the growth of predator population is densit-dependent, densit ma affect the actual mortalit or the actual birth rate, more likel affect both of them at the same time in different degree. Here, assume that onl the actual birth rate of predator population is affected b its densit. Assume that the carring capacit of predator population is a function of its pre densit, and denote it as φ(). Thus, according to model (1.1), one can formulate model (2.1). = r(1 ) α p + µ, = d + β p + (1 (2.1) φ() ). Here, > is the carring capacit of pre, and µ > is the loss rate of pre α that is irrelevant to predator. The predation rate is Holling II tpe, that is, p+, where α is the maimum predation rate, and predation rate is half maimal at = p. Parameter β is the maimum birth rate of predator. Assume that φ() is sufficientl smooth, and when, φ() > holds. Here, onl the isolated equilibriums of model (2.1) are considered. Theorem 2.1. Model (2.1) alwas has trivial equilibrium O(, ). When r > µ, model (2.1) has predator-onl equilibrium E( (r µ) r, ), and when r µ > rdp (β d) >, model (2.1) has positive equilibrium.

3 1466 H. Liu, T. Li & F. Zhang Proof. The eistence of trivial equilibrium and predator-onl equilibrium is obvious. Denote f() r (r µ) α ( + p)[ r ], g() ( β d β dp β )φ(), then the positive equilibrium of model (2.1) satisfies = f(), = g(). Obviousl, f( (r µ) r ) =, g( (r µ) r ) > and when r µ > rdp (β d) and g( dp ) = hold. So, there eists solution of f() = g() between dp β d (r µ) r dp >, f( β d ) > β d and, thus, model (2.1) has positive equilibrium. Through observing and analzing the images of function = f() and = g(), one can know that the positive equilibrium (, ) of model (2.1) ma fall into four classes: A, B, C, D, and the A class positive equilibrium must eist (Figure 1). In Figure 1, the downward parabola is image of = f(), and the rest curves are image of = g(). The characteristics of these four classes of positive equilibrium are: Class A: g ( ) > f ( ), Class B: g ( ) < f ( ), Class C: g ( ) = f ( ), g ( ) < f ( ), Class D: g ( ) = f ( ), g ( ) > f ( ). One can know that there ma eist positive equilibrium satisfing g ( ) = f ( ) and g ( ) = f ( ). This kind of positive equilibrium is similar to class C or class D and is omitted. Figure 1. Model (2.1) ma have four classes of positive equilibrium. Theorem 2.2. When r < µ, the trivial equilibrium O of model (2.1) is locall asmptoticall stable, otherwise, O is unstable. When < r µ < rdp (β d), the predator-onl equilibrium E of model (2.1) is locall asmptoticall stable, otherwise, E doesn t eist or is unstable. Proof. The Jacobian matri of model (2.1) at O is J O = r µ. d

4 A Pre-Predator model with Holling II functional response Its eigenvalues are r µ, d. So when r < µ, both eigenvalues are less than zero and O is locall asmptoticall stable. The Jacobian matri of model (2.1) at E is J E = µ r α(r µ) rp+(r µ) rdp+(β d)(r µ) rp+(r µ). Its eigenvalues are µ r, rdp+(β d)(r µ) rp+(r µ). So when < r µ < rdp (β d), both eigenvalues are less than zero and E is locall asmptoticall stable. Theorem 2.3. Denote h() 2r 2 + [rp + (β + µ r d)] pd. (i) When r µ > rdp (β d) >, h( ) >, the positive equilibrium of class A is locall asmptoticall stable, and the positive equilibrium of class B is a saddle. (ii) When r µ > rdp (β d) >, h( ) <, the positive equilibrium of class A is unstable, and the positive equilibrium of class B is a saddle. Proof. The Jacobian matri of model (2.1) at positive equilibrium P (, ) is J P = r + α α ( +p) 2 +p pd ( +p) + β ( ) 2 φ ( ) ( +p)φ 2 ( ) β ( +p)φ( ). The characteristic equation of J P is λ 2 + a 1 λ + a 2 =, where a 1 = h( ) ( +p), a 2 = αβ( ) 2 ( +p) 2 φ( ) (g ( ) f ( )). For the positive equilibrium of class A, a 2 > holds, and condition h( ) > implies a 1 >. So both eigenvalues have negative real parts and the positive equilibrium is locall asmptoticall stable. For the other cases, the proof is similar. Theorem 2.4. When h( ), the positive equilibrium of class C or D is a saddle-node. Proof. It is eas to know that the Jacobian matri of model (2.1) at class C or D positive equilibrium has one and onl one eigenvalue. = Making transformations dt = ( + p)φ()d τ, and ỹ = = α + αȳ, ỹ = (r µ rp 2r pd ) + (β d )ȳ, in turn, one can transform model (2.1) into model (2.2). At the same time, the positive equilibrium (, ) of model (2.1) changes to the trivial equilibrium O(, )

5 1468 H. Liu, T. Li & F. Zhang of model (2.2). d d τ = α 2 h( ) ( β φ ( ) 2φ( ) + pdφ ( ) dpφ( ) ( ) + rβ 2 α ) 2 [ h( ) ( α2 β φ ( ) φ( ) + 2α2 dpφ ( ) 2α2 dpφ( ) ( ) + 2rαβ 2 ) β (r µ rp 2r ) αβ ] ȳ { h( ) [ α2 β φ ( ) 2φ( ) (β d)α 2 φ ( ) αβ2 φ( ) + rαβ ] +αβ + αβ φ ( ) φ( ) β2 φ( ) }ȳ2 + o(ρ 2 ), dȳ d τ = φ( )h( ) ȳ + [ α2 h( ) ( β φ ( ) 2φ( ) + pdφ ( ) dpφ( ) ( ) 2 ) α( ) 2 φ( ) h( ) ( 2r + µ r + rp )] 2 +[(β α2 h( ) )( 2r + rp µ r) + h( ) ( α2 β φ ( ) φ( ) + 2α2 dpφ ( ) 2α2 dpφ( ) ( ) αβh( ) 2 +{ h( ) [ α2 β φ ( ) 2φ( ) + (β d)α 2 φ ( ) αβ2 φ( ) ] β2 φ( ) α h( ) ( 2r + rp r + µ)(h( )φ ( ) + r φ( ) + φ( )h( ) )}ȳ 2 +o(ρ 2 ). (2.2) Let L φ( )h( ), then transformation d τ = L d τ transforms model (2.2) into model (2.3). d d τ = a a 11 ȳ + a 2 ȳ 2 + o(ρ 2 ) Φ(, ȳ), (2.3) dȳ d τ = ȳ + b b 11 ȳ + b 2 ȳ 2 + o(ρ 2 ) ȳ + Ψ(, ȳ). Setting ȳ = c o( 2 ) and substituting it into ȳ + Ψ(, ȳ( )) =, one can get c 2 2 +b 2 2 +o( 2 ) =. Thus, c 2 = b 2, that is, ȳ = b 2 2 +o( 2 ). Substituting this relation into Φ(, ȳ( )), one can obtain Φ(, ȳ) = a o( 2 ). The fact that a 2 = α2 β 2 ( ) 2 2φ( )h 2 ( ) (g ( ) f ( )). implies that the trivial equilibrium O(, ) of model (2.2) is a saddle-node [3]. That is, the positive equilibrium of class C or D of model (2.1) is a saddle-node. Theorem 2.5. When h( ) =, the positive equilibrium of class A ma be a center or a focus, the positive equilibrium of class B is a saddle, the positive equilibrium of class C or D is a degenerate singular point. Proof. When h( ) =, the Jacobian matri of model (2.1) at the positive equilibrium P (, ) is J P = β ( +p)φ( ) β g ( ) ( +p)φ( ) α +p β ( +p)φ( ) Its characteristic equation is λ 2 = αβ( ) 2 (f ( ) g ( )) ( +p) 2 φ( ) Λ. For the positive equilibrium of class A, Λ < and J P has a pair of pure imaginar characteristic roots. Hence, the positive equilibrium ma be a center or a focus, one can use formal series method to analze furthermore. For the positive equilibrium of class B, Λ >. )] ȳ

6 A Pre-Predator model with Holling II functional response and J P has a positive and a negative eigenvalue, so the positive equilibrium is a saddle. For the positive equilibrium of class C or D, Λ = and J P has a double characteristic root. Making transformations dt = ( + p)φ()d τ, and =, ỹ =, = αφ( ) β + αφ( ) β 2 ( ) 2 ȳ, ỹ =, in turn, one can transform model (2.1) into model (2.4), and transform the positive equilibrium (, ) of model (2.1) into the trivial equilibrium Õ(, ) of model (2.4). d d τ = ȳ + αφ( )[ 1 + α(β d)φ( )φ ( ) β 2 + α φ ( ) 2β ] 2 + αφ( ) β [ 1 + 2α(β d)φ( )φ ( ) β 2 + α φ ( ) β + β α φ( ) ] ȳ + α2 φ 2 ( ) 2β 3 ( ) 2 ( 2φ( )φ ( ) β + φ ( ))ȳ 2 + o(ρ 2 ) ȳ + φ(, ȳ), dȳ d τ = α φ( )[ β + r φ( ) + α(β d)φ( )φ ( ) β + α φ ( ) 2 ] 2 +[αφ( ) + α φ( ) + α φ ( ) β 2rα φ 2 ( ) β 2α2 (β d)φ 2 ( )φ ( ) β α2 φ( )φ ( ) 2 β + αφ( ) β (1 + φ ( ) φ( ) r β αφ( )φ ( ) β 2 α φ ( ) 2β ) + o(ρ 2 ) ψ(, ȳ). Rewrite model (2.4) as model (2.5), d d τ = ȳ + A A 11 ȳ + A 2 ȳ 2 + o(ρ 2 ) ȳ + Ψ(, ȳ), dȳ d τ = B B 11 ȳ + B 2 ȳ 2 + o(ρ 2 ) Φ(, ȳ). ] ȳ (2.4) (2.5) Letting ȳ = a a and substituting it into ȳ + Ψ(, ȳ) =, one can obtain a 2 = A 2, a 3 = A 11 A 2, that is, ȳ = A A 11 A Using this relation, one can obtain Φ(, ȳ) = B o( 3 ), Ψ (, ȳ) + Φ ȳ(, ȳ) = (2A 2 +B 11 ) +o( 2 ) and B 2 = α2 ( ) 2 φ 2 ( ) 2. Hence, the positive equilibrium of class C or D of model (2.1) is degenerate singular point [3]. 3. Eamples and simulation In this section, we give some eamples of φ() and perform corresponding numerical simulation. In simulation, the parameter values are r = 1, = 15, α =.6, µ =.2, d =.1, β =.2, p = 5, and at this time, r µ =.98 > 1 3 = rdp (β d). In Figure 2, 3, 4, 5, the solid lines are the trajectories of model (2.1), the dashed lines are the images of = f() = 1 9 ( ) or = g() = ( )φ()( ), and

7 147 H. Liu, T. Li & F. Zhang the intersections of = f() and = g() are the positive equilibriums of model (2.1). If φ() = 2, the carring capacit of predator is constant. Model (2.1) has onl one positive equilibrium D, which is class A and is globall asmptoticall stable (Figure 2) D Figure 2. When φ() = 2, the positive equilibrium of model (2.1) and its stabilit. If φ() = 2 + 1, the carring capacit of predator increases linearl with the number of its pre increasing, model (2.1) has onl one positive equilibrium F, which is class A and is globall stable (Figure 3) F Figure 3. When φ() = 2 + 1, the positive equilibrium of model (2.1) and its stabilit. If φ() = ( 9) , model (2.1) has one positive equilibrium of class A, G1 and one positive equilibrium of class D, G2 (Figure 4). The positive equilibrium G2 is a saddle-node, its separatri (the dot line in Figure 4) divides the first quadrant into two parts, the smaller part is the attraction domain of G1, and the lager part is the attraction domain of G2. If φ() = 18 sin(/12.5) + 2, the carring capacit of predator changes with period 25π, model (2.1) has two positive equilibriums of class A, H1 and H3 and

8 A Pre-Predator model with Holling II functional response one positive equilibrium of class B, H2 (Figure 5). The two dot lines in Figure 5 are the trajectories that enter into saddle H2, the divide the first quadrant into two parts, and this two parts are attraction domain of H1 and H3 respectivel G G2 1 H1 5 5 H Figure 4. When φ() = ( 9) , the positive equilibriums of model (2.1) and their stabilit. H Figure 5. When φ() = 18 sin(/12.5) +2, the positive equilibriums of model (2.1) and their stabilit. When h( ) <, the positive equilibrium of class A is unstable, and there ma eist limit ccle around it (Figure 6). In Figure 6, the parameter values are r =.9, = 1, α =.75, µ =.2, d =.5, β = 5.5, p = 12 and φ() = 2. This time, the unique positive equilibrium is unstable, h( ) = <, and a limit ccle appears Q 5 1 Figure 6. When h( ) <, there eists limit ccle around class A positive equilibrium. 4. Application The alpine meadow is an important ecosstem in China that provides important ecological services. Over the past few decades, it has been observed that man alpine meadows have been degraded resulting in severe consequences in the ecolog,

9 1472 H. Liu, T. Li & F. Zhang environment and econom. In alpine meadow, plateau pika feeds on vegetation, that is, plateau pika is predator and vegetation is pre. At the same time, the carring capacit of plateau pika is influenced b the amount of vegetation also. So the relations between plateau pika and vegetation can be described b model (2.1). Net, we tr to eplore wh alpine meadow degraded and how recover it with the help of model (2.1). All parameters in model (2.1) are chosen on the basis of 1 ha of alpine meadow, and we use ear as the time unit. We take p = 2kg empiricall, and determine other parameter values basing on [12] and references therein. The values of r, d, are.227,.95, 4kg respectivel. One plateau pika consumes.36kg vegetation, α and assume that it happens at = 2kg, that is, p+ = α =.36, so we take α =.72. Similarl, the birth rate of plateau pika is 2.625, and assume that it happens at = 2kg, that is, = 2.625, so we take β = The β p+ = β carring capacit of plateau pika is φ() = (+1) (+1) which increases firstl 4 and then decreases with increasing. And φ() attains its maimum value 265 at = Other than plateau pika, vegetation ma reduce owing to grazing, and µ is the grazing rate. The images of = f() and = g() in Figure 7 shows that model (2.1) has onl one positive equilibrium which belongs to class A and is stable. When µ increases, curve = f() moves leftward and the positive equilibrium moves leftward along curve = g(). If µ is smaller, then with µ increasing the positive equilibrium moves left-upward, it results in vegetation decreasing and plateau pika increasing. This phenomenon is consistent with the realit of alpine meadow degrading. Hence, µ increasing, that is, overgrazing is the cause of alpine meadow degradation. When µ is larger, then with µ increasing the positive equilibrium moves left-downward, it results in both vegetation and plateau pika decreasing. This time, alpine meadow loss its function severel. When µ is greater than.2145 (namel the value of rdp (β d) ), the positive equilibrium disappears, plateau pika becomes etinct, vegetation is less, and the alpine meadow sstem is incomplete =f() µ=.1 =g() d=1.25 =f() µ=.5 =g() d= Figure 7. The influence of µ and d on positive equilibrium of (2.1). On the other hand, due to irrational hunting, the natural enemies of plateau pika decreased heavil, and the mortalit of plateau pika reduced. As shown in Figure 7, when d decreases, the curve = g() epands left-upward, and the positive

10 A Pre-Predator model with Holling II functional response equilibrium moves left-upward. This time, plateau pika increases and vegetation decreases. Hence, decreasing of plateau pika mortalit is also a cause of alpine meadow degrading. Up to now, we have known that both grazing rate increasing and plateau pika mortalit decreasing are the cause of alpine meadow degrading. Thus, we can recover the degraded alpine meadow through decreasing grazing rate and increasing plateau pika mortalit correspondingl. From Figure 7, one can know that decreasing µ and increasing d would result in the positive equilibrium moving rightdownward, that is, vegetation increasing, plateau pika decreasing and even becoming etinct. Although reducing grazing rate and increasing plateau pika mortalit seem to have the same effect, but the are different obviousl. Reducing grazing rate can increase vegetation significantl, even near to its carring capacit. Whereas, through increasing plateau pika mortalit, the recover of vegetation is limited. When µ =.1, vegetation onl can recover to 26kg approimatel through increasing d. Therefore, decreasing grazing rate is the fundamental strateg to recover degraded alpine meadow. 5. Discussion Because of the complicated relations in ecological sstem, the carring capacit of predator ma be determined b the amount of its pre. Considering this phenomenon, this paper formulated a pre-predator model with Holling II functional response and analzed its global behavior. This model ma have positive equilibrium or limit ccle, and the positive equilibrium ma be stable, or a saddle-node, or a saddle, or a degenerate singular point. The global behavior of this model ma alter with the relation between carring capacit of predator and amount of its pre changing. Thus, before studing or managing populations, it is needed to understand the relations among populations full. References [1] A. Berrman, The orgins and evolution of predator-pre theor, Ecolog, 1992, 73(5), [2] Y. Chu and C. Ding, Pre-predator sstems with intertwined basins of attraction, Applied Mathematics and Computation, 21, 215(12), [3] S. Cai and X. Qian, Introduction To Qualitative Theor Of Ordinar Differential Equations, Higher Education Press, Beijing, [4] M. Falcone and G. Israel, Qualitative and numerical analsis of a class of prepredators models, Acta Applicandae Mathematica, 1985, 4(2 3), [5] M. Falcone, G. Israel and L. Tedeschini-Lalli, A Class Of Pre-predator Models With Competition Between Predators, Sapienza Universit of Rome, Rome, [6] C. Holling, The functional response of predators to pre densit and its role in mimicr and population regulation, Memoirs of the Entomological Societ of Canada, 1965, 97(45), 5 6.

11 1474 H. Liu, T. Li & F. Zhang [7] M. A. Idlango, J. J. Shepherd and J. A. Gear, Logistic growth with a slowl varing Holling tpe II harvesting term, Communications in Nonlinear Science and Numerical Simulation, 217, 49, [8] Z. Jing, Qualitative analsis for dnamic sstem of predator-pre, Acta Mathematicae Applicatae Sinica, 1989, 12(3), [9] R. ooij and A. Zegeling, A predator-pre model with Ivlev s functional response, Journal of Mathematical Analsis and Applications, 1996, 198(2), [1] H. Liu, The Simulation Spatio-temporal Dnamics Of The Plateau Pika Population Using A Cellular-autimata Model, Graduate School of Chinese Academ of Sciences, Beijing, 28. [11] H. Li and Y. Takeuchi, Dnamics of the densit dependent predator-pre sstem with Beddington-DeAngelis functional response, Journal of Mathematical Analsis and Applications, 211, 374(2), [12] H. Liu, F. Zhang and Q. Li, A pre-predator model in which the carring capacit of predator depending on pre, Mathematica Applicata, 217, 3(4), [13] L. Tedeschini-Lalli, Smoothl intertwined basins of attraction in a pre-predator model, Acta Applicandae Mathematica, 1995, 38(2), [14] H. Tang and Z. Liu, Hopf bifurcation for a predator-pre model with age structure, Applied Mathematical Modelling, 216, 4(2), [15] P. Tu and E. Wilman, A generalized predator-pre model: uncertaint and management, Journal of Environmental Economics and Management, 1992, 23(2), [16] X. Wang and Y. Wang, Novel dnamics of a predator-pre sstem with harvesting of the predator guided b its population, Applied Mathematical Modelling, 217, 42, [17] J. Yang and S. Tang, Holling tpe II predator-pre model with nonlinear pulse as state-dependent feedback control, Journal of Computational and Applied Mathematics, 216, 291, [18] J. Zhang and Z. Ma, Qualitative analsis to a class of pre-predator sstem with Holling s tpe II functional response and having two limit ccle, Journal of Biomathematics, 1996, 11(4), [19] J. Zhang and Z. Ma, The uniqueness of limit ccles of a predator-pre sstem with Holling s tpe II functional response and dependent densit of predators and pres, Journal of Biomathematics, 1996, 11(2), 26 3.