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1 Nonlinear Analsis: Real World Applications ( Contents lists available at ScienceDirect Nonlinear Analsis: Real World Applications journal homepage: Qualitative analsis of a predator pre model with constant-rate pre harvesting incorporating a constant pre refuge Lili Ji, Chengqiang Wu College of Mathematics and Computer Science, Fuzhou Universit, Fuzhou, Fujian, 3000, PR China a r t i c l e i n f o a b s t r a c t Article histor: Received Ma 009 Accepted Jul 009 Kewords: Predator pre model Pre refuge Limit ccle Harvesting Global stabilit We consider a predator pre model with Holling tpe II functional response incorporating a constant pre refuge and a constant-rate pre harvesting. Depending on the constant pre refuge m, which provides a condition for protecting m of pre from predation, and the constant-rate pre harvesting, some sufficient conditions for the instabilit and global stabilit of the equilibria, and the eistence and uniqueness of limit ccles of the model are obtained. We also show the influences of pre refuge and harvesting efforts on equilibrium densit values. Numerical simulations are carried out to illustrate the feasibilit of the obtained results and the dependence of the namic behavior on the harvesting efforts or pre refuge. 009 Elsevier Ltd. All rights reserved.. Introduction The stu of the consequences of hiding behavior of pre on the namics of predator pre interactions can be recognized as a major issue in applied mathematics and theoretical ecolog [ 3]. Some of the empirical and theoretical work have investigated the effect of pre refuges and drawn a conclusion that the refuges used b pre have a stabilizing effect on the considered interactions and pre etinction can be prevented b the addition of refuges [ ]. Gonzalez-Olivares and Ramos-Jiliberto [9] studied the namic consequences of the following predator pre sstems with constant number of pre using refuges, which protects m of pre from predation: = α ( k β( m + a( m, = + cβ( m + a( m. The showed the effects of the two models of refuge on the namic propert, such as the local stabilit of equilibria and the eistence of limit ccle. Chen [0] gave the complete qualitative analsis for model (.. Motivated b the paper of Gonzalez-Olivares and Ramos-Jiliberto [9], we consider the following predator pre model with Holling tpe II functional response incorporating a positive constant pre refuge and a nonzero constant-rate pre harvesting = α ( k = + cβ( m + a( m, β( m + a( m h, (. (. Corresponding author. addresses: lilms@3.com (L. Ji, cqw00@fzu.edu.cn (C. Wu. -/$ see front matter 009 Elsevier Ltd. All rights reserved. doi:0.0/j.nonrwa

2 L. Ji, C. Wu / Nonlinear Analsis: Real World Applications ( where and denote the pre and predator population sizes, respectivel, at an time t; α > 0 represents the intrinsic growth rate of the pre; k is the carring capacit of the pre in the absence of predator and harvesting; m is a constant number of pre using refuges, which protects m of pre from predation; the term β/(+a denotes the functional response of the predator, which is known as Holling tpe II response function []; c > 0 is the conversion factor denoting the number of newl born predators for each captured pre; d > 0 is the death rate of the predator; h > 0 is the rate of pre harvesting. From the point of view of human needs, the eploitation of biological resources and the harvest of population are commonl practiced in fisher, forestr and wildlife management. Concerning the conservation for the long-term benefits of humanit, there is a wide range of interest in the use of bioeconomic modeling to gain insight into the scientific management of renewable resources like fisheries and forestries. The problem of predator pre interactions under constant rate of harvesting of either species or both species simultaneousl have been studied b some authors. For eample, Brauer and Soudak [ ] and Xiao [,] studied a class of predator pre models under constant rate of harvesting and under constant quota of harvesting of both species simultaneousl. The showed how to classif the possibilities of the quantitative behavior of the solutions to locate the set of initial values in which the trajectories of the solutions approach to either an asmptotic stable equilibrium or an asmptoticall stable limit ccle. Recentl, Dai and Tang [9] studied a predator pre model in which two ecological interacting species are harvested independentl with constant rates. The showed that the harvested predator pre sstem ma ehibit ver complicated namics such as a spontaneous appearance of a homoclinic orbit and multiple limit ccles. The rest of the paper is organized as follows. Basic properties such as the eistence, stabilit and instabilit of the equilibria of the model are given in Section. In Section 3, sufficient conditions for the global stabilit of the unique positive equilibrium are obtained. In Section, we derive the eistence and uniqueness of limit ccle. In Section, we analze the influence of pre refuge and harvesting efforts and give numerical stimulations. This paper ends b a brief conclusion.. Basic properties of the model Let Ω 0 = {(, > m, > 0}. For practical biological meaning, we simpl stu sstem (. in Ω 0 or in Ω 0. From the first equation of sstem (., it is eas to derive lim sup t + (t k. Lemma.. The solution ((t, (t of sstem (. with the initial values (0 > m, (0 > 0 is positive and bounded for all t 0. Proof. Obviousl, the solution ((t, (t of sstem (. with the initial values (0 > m, (0 > 0 is positive for all t 0. Define the function ω(, = c +. Given an ε > 0, (t k + ε for t sufficientl large. From sstem (., it follows that ( ( dω = cα k ch cα k then ω cα(k+ε, for t sufficientl large, which completes the proof. min{d,α} Lemma. ([0]. Assume that stable (unstable limit ccle. = X(,, min{d, α}ω + cα(k + ε, = Y(,. Along a limit ccle L, if div(x, Y < 0(> 0 holds, then L is a L Lemma.3 ([0]. Assume that = X(,, ε, = Y(,, ε, X(,, ε, Y(,, ε structures a rotation vector field in the wider sense. And furthermore suppose that the set of equations admit a semi-stable limit ccle L ε0 for ε = ε 0. Then, when parameter ε changes according to appropriate direction, L ε0 at least decomposes to a stable and an unstable limit ccle, and the la in inboard and outboard of L ε0, respectivel; while ε changes in opposite direction, L ε0 disappears. For simplicit, we take the following scaling: = m, = β, = ( + a( mdτ and then sstem (. takes the following form (still denote,, τ as,, t = b( + a[ + (k m + km m u] P(,, = (cβ a Q (,, where b = α k, u = h b. Clearl, Ω 0 transforms into Ω = (, > 0, > 0 and sstem (. is bounded. Considering the function Φ( = + (k m + km m u, the discriminant of Φ( = 0 takes the following form = (k m + (km m u = k u. If = k u > 0 and Φ(0 = km m u > 0 hold, Φ( has onl one positive root 0 = k m+ k u. (.

3 L. Ji, C. Wu / Nonlinear Analsis: Real World Applications ( 3 So, if 0 < u < km m holds, sstem (. has at most two equilibria P 0 ( 0, 0, P (,, where d = cβ ad, = b( + a Φ(. P (, is a unique positive equilibrium if and onl if cβ ad > 0, 0 < m < k+ k u d. cβ ad In what follows, we onl consider the case that 0 < u < km m. Lemma.. If m > k+ k u d cβ ad holds, then P 0 is a stable node point; if 0 < m < k+ k u d cβ ad holds, then P 0 is a saddle point. Furthermore assume that m = [(k + a ( + a + k u]/, then (I P is locall asmptoticall stable for all 0 < m < k+ k u d, if ak ad3 u(cβ ad 3 holds; cβ ad (cβ add (II P is locall asmptoticall stable for m < m < k+ k u d and P cβ ad is locall unstable for 0 < m < m, if ak > ad3 u(cβ ad 3 holds. (cβ add Proof. The Jacobian matri of sstem (. for the equilibrium point P 0 is given b ( ρ J(P 0 = 0, 0 ρ where ρ = b( + a 0 ( 0 + k m and ρ = (cβ ad 0 d. If m > k+ k u d holds, the eigenvalues cβ ad of matri J(P 0 are ρ, ρ which are negative, hence, P 0 is locall asmptoticall stable and furthermore it is a stable node point. If 0 < m < k+ k u d holds, one of the eigenvalues of matri J(P cβ ad 0 is ρ which is positive, hence, P 0 is locall unstable and furthermore P 0 it is a saddle point. The Jacobian matri of sstem (. for P is given b ( P J(P =, ω 0 where P = abφ( + b( + a φ ( and ω = cβ ad. Note that tr(j(p = b [m (k + a m ( ak a 3 + u]. Considering the function R( = (k + a ( ak a 3 + u, the discriminant of R( = 0 takes the following form = (k + a + [( ak a 3 + u] = + k u > 0, therefore, R( has two roots m = [(k + a ( + a + k u]/, m = [(k + a + ( + a + k u]/. It is eas to see that m > k. (I If ak ad3 u(cβ ad 3 (cβ add holds, then m 0 and for all m < < m, R( < 0, hence, for all 0 < m < k+ k u d cβ ad < 0, P is locall asmptoticall stable. (II If ak > ad3 u(cβ ad 3 holds, then m (cβ add > 0 and R( < 0 for all m < < m, R( > 0 for all 0 < < m. Notice that > ( + a, then m < (k+a (+a < k+ k u d, hence, if m cβ ad < m < k+ k u d holds, cβ ad then tr(j(p < 0 and P is locall asmptoticall stable; if 0 < m < m holds, then tr(j(p > 0 and P is locall unstable. This completes the proof. Theorem.. If m > k+ k u d cβ ad hold, then P 0 is globall asmptoticall stable. Proof. Notice that m > k+ k u d.p cβ ad 0( 0, 0 is the unique equilibrium of sstem (.. If sstem (. eists a closed orbit in Ω, then there must eist an equilibrium in the interior of the closed orbit, which is impossible. Hence, sstem (. does not eist limit ccle for all m > k+ k u d. From the boundedness of sstem (., the stable node point P cβ ad 0 is globall asmptoticall stable.

4 L. Ji, C. Wu / Nonlinear Analsis: Real World Applications ( 3. Global stabilit for the positive equilibrium Theorem 3.. Assume that 0 < u < km m, and one of the following conditions holds: (H ak min{ ad3 u(cβ ad 3, am}, 0 < m < k+ k u d ; (cβ add cβ ad (H ad3 u(cβ ad 3 < ak am, m (cβ add < m < k+ k u d. cβ ad Then the positive equilibrium P (, of sstem (. is globall asmptoticall stable. Proof. Define a Dulac function B(, =, from sstem (., we have D = (BP + (BQ = b [a 3 + ( ak + am + km m u] b ϕ(. Then ϕ ( = [3a + ( ak + am] = 0 has two roots = 0, ak + am =. 3a If ak am, then 0. For all 0, ϕ ( 0 and ϕ ( = 0 holds if and onl if = 0. Hence, ϕ( is increasing in [0, +. From ϕ(0 = km m u > 0, we derive ϕ( > 0, for all > 0, therefore, for all 0, D < 0, sstem (. does not eist limit ccle. If one of (H, (H holds, P is locall asmptoticall stable and sstem (. does not eist limit ccle in Ω. From the boundedness of sstem (., P is globall asmptoticall stable.. Eistence and uniqueness of limit ccle Theorem.. Suppose that 0 < u < km m, ak > ad3 u(cβ ad 3, and 0 < m < m. Sstem (. admits at least one limit ccle in Ω. (cβ add Proof. For sstem (., constructing a Bendison ring OA BCD including P. Define OA as a length of line L = = 0, AB as a length of line L = k m+ k u = 0. Define = (a 0, = [ d + (cβ ad], where a 0 = ma 0 b( + a[ + (k m + km m u]/. The orbit of sstem (. with the initial value point B( k m+ k u, a 0 intersects with line = and the intersection point is C(,, then we derive the curve BC. Define CD as a length of line L 3 = = 0, DO as a length of line L = = 0. Since dl (. = k m+ k u < 0( > 0, dl 3 (. = (cβ a < 0(0 < <, dl (. = b(km m u > 0, the orbits of sstem (. go through into the interior of the Bendison ring from the outer of AB, CD and DO; on BC, compared sstem (. with sstem (., we have (. < (. < 0 and (. = (. > 0, then the orbits of sstem (. go through into the interior of the Bendison ring from the outer of BC. On the other hand, under the assumptions of Theorem., P (, is an unstable equilibrium. B Poincare Bendison Theorem, Sstem (. admits at least one limit ccle in region OA BCD Ω. This completes the proof of Theorem.. Theorem.. Suppose that 0 < u < km m, ak > ad3 u(cβ ad 3, and 0 < m < m. Sstem (. admits at most one limit ccle which is globall asmptoticall stable in Ω. (cβ add Proof. In order to prove Theorem., we take the following change of variables v =, u = ln, dτ =, still denote v, u, τ as,, t, then sstem (. takes the following form = b( + a[ + (k m + km m u] e +, d + (cβ ad =, and the positive equilibrium P (, P (, ln. (. (.

5 L. Ji, C. Wu / Nonlinear Analsis: Real World Applications ( Taking translation transformation = +, = + ln, then P transform to the origin O(0, 0, still denote, as,, then sstem (. takes the following form where = e + b( + a( + [ ( + + (k m( + + km m u] + φ( F(( >, = d + (cβ ad( + g(, + F( = b( + a( + [ ( + + (k m( + + km m u] +, f ( = F ( = b[a( ( ak + am( + + km m u] ( +. Notice that under the assumptions of Theorem., P is an unstable equilibrium and tr(j(p > 0, thus f (0 = tr(j(p < 0 and g(0 = 0, when 0, g( > 0. Under the assumptions of Theorem., O(0, 0 is an unstable equilibrium of sstem (.3. Thus, if there eist closed orbits of sstem (.3, then define L as the nearest orbit to the equilibrium O and L is inboard stable. Define the parameter equation of the closed orbit L : = (t, = (t. B Lemma., [ ( φ( F( div(x, Y = L L = F ( L = f ( (t < 0, L + (g( then L f ( (t 0. Furthermore suppose that there eists a closed orbit L L of sstem (.3, where L is net to L and the parameter equation of the closed orbit L is = (t, = (t. Thereinafter, we depart the proof of Theorem. to two steps: (I f ( L (t > f ( L (t; (II L is impossible to be a semi-stable limit ccle. Therefore, if (II holds, then L must be stable; b (I, there eists no other closed orbits ecept L, which completes the proof of uniqueness. ] (.3 Proof (I. Constructing a function f ( = f ( + δg(. Suppose that Q, P are the least and most -coordinate points in L respectivel, and the must la on the curve φ( F( = 0. Let δ = f ( Q g( Q, then f (0 = f (0 < 0, f ( Q = 0. Doing a perpendicular line passing through Q, the perpendicular line intersects with -ais and the foot of a perpendicular M, then f ( M = f ( Q = 0. We shall show: (T f ( must intersect with positive -semi-ais and the -coordinate of the intersection point N satisfing N P ; (T For all ( M, N, f ( < 0; (T 3 For all (, M ( N, +, function f ( is non-decreasing. g( Proof (T. If (T does not hold, then on region L, f ( 0, and therefore f ( (t = f ( (t δ g( (t L L L = f ( (t δ L L = f ( (t < 0, L which contradicts f ( L (t 0. This completes the proof of (T. From the proof of (T, we derive f L ( i (t i > f ( L i i (t, i =,. Thus, (I can be rewrite as the following form (I f ( (t > f ( (t. L L

6 L. Ji, C. Wu / Nonlinear Analsis: Real World Applications ( Proof (T. f ( = f ( + δg ( { } b[a( + 3 (km m u] = + δd ( + + ψ( ( +, ψ ( = b[a( (km m u] > 0, >. + Clearl, ψ( is increasing in (, +. If ψ( 0, then f ( 0, which contradicts f ( M = 0, f (0 < 0; if ψ( 0 then f ( 0, which contradicts f (0 < 0, f ( N = 0. Hence, there must eist a 0 ( M, N so that ψ( 0 = 0 and < 0, ψ( < 0, > 0, ψ( > 0, then f ( 0 = 0 and < 0, f ( < 0, > 0, f ( > 0. These combined with f ( M = f ( N = 0, impl that for all ( M, N, f ( < 0. Proof (T 3. where f ( g( = f ( g( bψ ( [(cβ ad( + d]( +, Ψ ( = (cβ ad( + [a( + 3 (km m u] d[a( ( ak + am( + (km m u]. From tr(j(p > 0 and d = (cβ ad, we derive Ψ ( = (cβ ad[a( + 3 a ( + ( ak + am( + (km m u], Ψ ( = (cβ ad[a( + ( ak + am]. Since tr(j(p > 0, ak + am < 0 and for all >, Ψ ( > 0, so, Ψ ( is increasing for all >. Since Ψ ( = (cβ ad(km m u < 0, Ψ (0 = (cβ ad[ a 3 ( ak + am (km m u] > 0, there eists a 0 (, 0 such that Ψ ( 0 = 0, where Ψ ( 0 = (cβ ad[a( a ( 0 + ( ak + am( 0 + (km m u] = 0, and hence, when < < 0, Ψ ( < 0; when > 0, Ψ ( > 0, so Ψ ( takes the least value at = 0. Notice that 0 < 0 and then a( + 3 a ( + (km m u = ( ak + am( +, Ψ ( 0 = (cβ ad( + [a( + 3 (km m u] (cβ ad [a( ( ak + am( + (km m u] = (cβ ad(m km + u + (cβ ad( + [a( (m km + u a ( + ( ak + am( + ] = (cβ ad(m km + u 0 a 0 (cβ ad( + 3 > 0. We derive Ψ ( > 0 for all >, which means for all (, M ( N, +, the function f ( is non-decreasing. B (T, (T, (T 3 and Green formula (the regions G, G, G 3, G refer to the figure in Zhang et al. [0], page, we have f ( (t f ( (t = ( ( f ( f ( + L L G G 3 φ( F( G G g( = f ( ( e (φ( + F( + f ( > 0, g( which completes the proof of (I. G G 3 Proof. (II. Assume that L is a semi-stable limit ccle and line = 3 < 0 intersects with L. Constructing a new function F( = F( εr(, where 0 < ε, { 0, 3, r( = (g( g( 3, < 3. G G g(

7 L. Ji, C. Wu / Nonlinear Analsis: Real World Applications ( It is obvious that (I f ( > L L f (, holds, where f ( = F (. On the other hand, for the parameters 0 < ε < ε, we have φ( F( + ε r( g( φ( F( + ε r( g( = r(g((ε ε 0. Therefore, ( φ( F(, g( structures a rotation vector field in the wider sense on parameter ε. Considering = φ( F(, = g(, (. when ε = 0, sstem (. is transformed into sstem (.3. B Lemma.3, when 0 < ε, the semi-stable limit ccle L of sstem (.3 decomposes into at least two rings L L of sstem (. and L is inboard stable, L is outboard unstable, then f ( 0, f ( 0, L L which contradicts (I. This completes the proof of (II and then Theorem. follows.. The influence of pre refuge and harvesting efforts and simulations I. The influence of pre refuge on model (. Notice that the scaling from sstem (. to sstem (.: = m, = β, = (+a( mdτ, then the coordinates of the positive equilibrium P (, of sstem (. take = + m, and = cα (k ch kd d = cα ( ( d d m + k m ch kd cβ ad cβ ad d. d cβ ad Notice that is a continuous differential function of parameter m and dm = > 0. The above inequalit shows that is the strictl increasing function of parameter m and that increasing the pre refuge leads to the increasing of the densit of pre species. One could see that is also a continuous differential function of parameter m. Simple computation shows that dm = cα kd ( k m d cβ ad. We will discuss (. in two cases: Case : Assume that the inequalit k d 0 holds, then < 0 for all m > 0, thus cβ ad dm is the strictl decreasing function of parameter m. That is, increasing the amount of pre refuge can decrease the densit of predator species. In this cα case, reaches the maimum value k(cβ ad (k d ch at m = 0. cβ ad d Case : Assume that the inequalit k d > 0 holds, denoting cβ ad m = k d, then it follows that > 0 for all cβ ad dm m (0, m and < 0 for all m dm (m, k. Thus, there eists a threshold m = m such that is the strictl increasing function of parameter m for all m (0, m, while is the strictl decreasing function of parameter m for all m (m, k. attains its maimum value c(kα h at m = m. d The above analsis shows that in the interval (0, m, increasing the amount of pre refuge can increase the densit of predator species and this is happened due to predator species still have enough food for predation with m being small, while as m large than m and the pre refuge becomes large enough, then increasing the amount of pre refuge can decrease the densit of predator species and this is happened due to the loss of food for predator species. Letα =, k = 0, d =, a = c = 0., β =, h =. B simple computation, we have m =., u =, m = 0, k+ k u d = 9.993, ak =, ad3 u(cβ ad 3 =.. Then cβ ad (cβ add (. (.

8 L. Ji, C. Wu / Nonlinear Analsis: Real World Applications ( Fig.. There is a stable limit ccle surrounding P (,. with m = Fig.. P (0,.33 is globall asmptoticall stable with m = 0. ( If 0 < m <., the conditions of Theorems. and. hold, so model (. admits eactl one limit ccle; ( If. < m < 9.999, (H holds, then P is globall stable; (3 If < m < 0, the conditions of Theorem. hold, so P 0 is globall stable. Figs. show that the densit of pre species is increasing while m is increasing and Figs. and show that the densit of predator species increase as m < 0 and Figs. 3 and show that the densit of predator species decrease as m > 0. Figs. also show that the dependence of the namic behavior of sstem (. on the pre refuge m. When m is small, there is a stable limit ccle surrounding the unique positive equilibrium, and when m is large enough, the limit ccle is broken and both the pre and predator population converge to their equilibrium values respectivel, which means that if we change the value of m, it is possible to prevent the cclic behavior of the predator pre sstem and to drive it to a required stable state. II. The influence of harvesting efforts on model (. We will discuss this for an fied parameter m. It is eas to calculate that: dh = c d < 0. The above analsis shows that increasing the harvesting effort h can decrease predator densit, which is obvious, because the increasing of h must lead to the loss of food for predator species. Furthermore, for fied m, we notice that the densit of pre species still have no change as t + and alwas keep the value +m, though increase harvesting pre species. Obviousl, the decreasing of densit of predator species can lead to the final immovabilit of densit of pre species. Let α =, k = 0, d =, a = c = 0., β =, m =. Figs. and shows that, the densities of pre species keep unchangeable when there eists the positive equilibrium, while the densities of predator species decrease with an increase in h. d cβ ad (.3

9 L. Ji, C. Wu / Nonlinear Analsis: Real World Applications ( Fig. 3. P (3,. is globall asmptoticall stable with m = Fig.. P (3,.9 is globall asmptoticall stable with m = Fig.. P (,. is globall asmptoticall stable with h =.. Conclusion In this paper we consider a predator pre model with Holling tpe II functional response incorporating a constant pre refuge and constant-rate pre harvesting. When the assumption 0 < h < α(km m /k holds, we show that the constant-

10 0 L. Ji, C. Wu / Nonlinear Analsis: Real World Applications ( Fig.. There is a stable limit ccle surrounding P (, 3. with h = 0. rate pre harvesting has no influence on the limit ccle. The sstem also admits a unique stable limit ccle when the positive equilibrium is unstable. Differentl from the conclusion of Chen [0] increasing the amount of refuge can increase pre densities and lead to population outbreaks, this paper verifies that increasing the amount of refuge can increase pre densities, and when the assumption k d 0 holds, increasing the amount of pre refuge can decrease the predator densities; when the assumption k cβ ad d cβ ad > 0 holds, there eists a threshold m, such that for the pre refuge smaller than this threshold, increasing the amount of pre refuge can increase the predator densities and if the pre refuge is larger than the threshold, increasing the amount of pre refuge can decrease the predator densities. The analsis of the influence of harvesting effort on equilibrium densit values indicates that for fied pre refuge, harvesting has no influence on the final densit of the pre species, while the densit of predator species is decreasing with the increasing of harvesting effort on pre species. Our results and numerical simulation also indicate that namic behavior of the model ver much depends on the pre refuge parameter m or harvesting effort h. Hence, it is possible to control the sstem in such a wa that the sstem approaches a required state, using the effort h or pre refuge m as controls. Acknowledgements This work is supported b the Natural Science Foundation of Fujian Province (00J009, and the Foundation of Science and Technolog of Fuzhou Universit (00-XQ-0. References [] C.S. Holling, Some characteristics of simple 0 tpes of predation and parasitism, The Canada Entomological 9 ( [] M.P. Hassell, The Dnamics of Arthropod Predator Pre Sstems, Princeton Univ. Press, Princeton, NJ, 9. [3] M.A. Ho, Almonds (California, in: W. Helle, M.W. Sabelis (Eds., Spider Mites: Their Biolog, Natural Enemies and Control, World Crop Pests, vol. B, Elsevier, Amsterdam, 9. [] A. Sih, Pre refuges and predator pre stabilit, Theoretical Population Biolog 3 (9. [] V. Krivan, Effects of optimal antipredator behavior of pre on predator pre namics:the role of refuges, Theoretical Population Biolog 3 (99 3. [] T.K. Kar, Stabilit analsis of a pre predator model incorporating a pre refuge, Communications in Nonlinear Science and Numerical Simulation 0 (00 9. [] W. Ko, K. Ru, Qualitative analsis of a predator pre model with Holling tpe II functional response incorporating a pre refuge, Journal of Differential Equations 3 ( [] J.B. Collings, Bifurcation and stabilit analsis of a temperature-dependent mite predator pre interaction model incorporating a pre refuge, Bulletin of Mathematical Biolog (99 3. [9] E. Gonzalez-Olivares, R. Ramos-Jiliberto, Dnamic consequences of pre refuges in a simple model sstem: More pre, fewer predators and enhanced stabilit, Ecological Modelling ( [0] L. Chen, F. Chen, Qualitative analsis of a predator pre model with Holling tpe II functional response incorporating a constant pre refuge, Nonlinear Analsis: Real World Applications (00 doi:0.0/j.nonrwa [] C.S. Holling, The functional response of predators to pre densit and its role in mimicr and population regulations, Memoirs of the Entomological Societ of Canada ( [] F. Brauer, A.C. Soudack, Stabilit regions and transition phenomena for harvested predator pre sstems, Journal of Mathematical Biolog ( [3] F. Brauer, A.C. Soudack, Stabilit regions in predator pre sstems with constant rate pre harvesting, Journal of Mathematical Biolog (99. [] F. Brauer, A.C. Soudack, Constant-rate stocking of predator pre sstems, Journal of Mathematical Biolog (9. [] F. Brauer, A.C. Soudack, Coeistence properties of some predator pre sstems under constant rate harvesting and stocking, Journal of Mathematical Biolog (9 0. [] F. Brauer, A.C. Soudack, On constant effort harvesting and stocking in a class of predator pre sstems, Journal of Theoretical Biolog 9 ( (9.

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