WIXLIB

International Journal of Ecology3where𝐴 𝑎11 ,𝐶1 𝑏11 ,𝐶2 𝑐22 ,𝐷2 𝑎11 𝑐22 𝑎12 𝑐21 ,𝐸 𝑏11 𝑐22 .Let(H4 )2(𝐴 𝐶2 ) 𝐶12 2𝐷2 0,(8)22[(𝐴 𝐶2 ) 𝐶12 2𝐷2 ] 4 (𝐷22 𝐸2 ) .I now discuss the following cases.𝜆2 (𝐴 𝐶1 𝐶2 ) 𝜆 (𝐷2 𝐸) 0.(9)𝜏̃1 𝑞 All roots of (9) will have negative real parts if and only if2𝐷2 𝐸 0.(10)Theorem 1. The interior equilibrium 𝐸 (𝑥 , 𝑦 ) of the system(3) exists and becomes locally asymptotically stable in absenceof delays if 𝑚 1 𝑟(𝑎 ℎ)/𝛼, 𝛼 𝑟(𝑎 ℎ) and (𝐻1 ) hold.2𝑞𝜋, ̃ 𝜔𝜆 (𝐴 𝐶2 ) 𝜆 𝐷2 (𝐶1 𝜆 𝐸) 𝑒 0.[𝑞 0, 1, 2, . . . .𝑑(Re(𝜆(𝜏1 )))]𝑑𝜏1𝜏̃1 ,𝜔 𝜔̃0(11)Let 𝑖𝜔 (𝜔 0) be a root of (11). Then I have 0̃04 𝐸2 𝐷22𝜔̃02 (𝐶12 𝜔̃02 𝐸2 )𝜔̃2𝜔 2 2 0 2 0,̃0 𝐸𝐶1 𝜔𝐸 cos 𝜔𝜏1 𝐶1 𝜔 sin 𝜔𝜏1 𝜔 𝐷2 ,𝐶1 𝜔 cos 𝜔𝜏1 𝐸 sin 𝜔𝜏1 (𝐴 𝐶2 ) 𝜔.(12)[𝑑Re(𝜆(𝜏1 ))] 0,𝑑𝜏1̃ 𝜏1 ̃𝜏1 ,𝜔 𝜔𝑞This leads to2𝜔4 [𝐶12 (𝐴 𝐶2 ) 2𝐷2 ] 𝜔2 𝐷22 𝐸2 0.𝐷22 𝐸2 0.(14)Hence, all roots of (13) will have negative real parts when𝜏1 [0, ) if conditions of Theorem 1 and (H2 ) are satisfied.Let(H3 )𝐷22 𝐸2 0.(15)If Theorem 1 and (H3 ) hold, then (13) has a unique positivẽ02 . Substituting 𝜔02 into (12), I haveroot 𝜔2𝜏̃1𝑝 2̃0 𝐷2 ) (𝐴 𝐶2 ) 𝐶1 𝜔̃0𝐸 (𝜔1cos 1 []222̃0̃0 𝐸𝜔𝐶1 𝜔2𝑝𝜋 ,̃0𝜔𝑝 0, 1, 2, . . . ,̃0 is the unique positive root of (13).where 𝜔𝑑[Re(𝜆(𝜏1 ))] 0.𝑑𝜏1̃ 𝜏1 ̃𝜏1 ,𝜔 𝜔(13)It follows that (12) has no positive roots if the followingconditions are satisfied:(H2 )2(19)by hypothesis (H4 ).Similarly, I can show that2(𝐴 𝐶2 ) 𝐶12 2𝐷2 0,(18) If 𝜆(𝜏1 ) is a root of (11) satisfying Re(𝜆(𝜏1𝑛 ))0 (or Re(𝜆(𝜏1 𝑞 )) 0, resp.) and Im(𝜆(𝜏1𝑛 )) 𝜔0 (or̃ , resp.), I obtainIm(𝜆(𝜏1 𝑞 )) 𝜔Case 2 (𝜏1 ̸ 0, 𝜏2 0). In this case, the characteristicequation (7) becomes 𝜆𝜏12̃ 𝐷2 ) (𝐴 𝐶2 ) 𝐶1 𝜔̃ 𝐸 (𝜔1cos 1 []222̃ ̃ 𝐸𝜔𝐶1 𝜔(H1 )2(17)̃ 2 andIf (H1 ) and (H4 ) hold, then (13) has two positive roots 𝜔2̃ into (12), we obtaiñ . Substituting 𝜔𝜔Case 1 (𝜏1 0 𝜏2 ). In this case, (7) becomes𝐴 𝐶1 𝐶2 0,𝐷22 𝐸2 0,𝑞Theorem 2. Assume that 𝜏1 ̸ 0, 𝜏2 0 and conditions ofTheorem 1 are satisfied, then the equilibrium 𝐸 (𝑥 , 𝑦 ) islocally asymptotically stable for 𝜏1 𝜏̃10 and unstable for 𝜏1 𝜏̃10 . Furthermore, the system (3) undergoes a Hopf bifurcationat (𝑥 , 𝑦 ) when 𝜏1 𝜏̃10 , where𝜏̃10 ̃02 𝐷2 ) (𝐴 𝐶2 ) 𝐶1 𝜔̃02𝐸 (𝜔1cos 1 [].̃0̃02 𝐸2𝜔𝐶12 𝜔(21)Case 3 (𝜏1 (0, 𝜏̃10 ), 𝜏2 ̸ 0). In this case, I allow a gestationtime period for the predator and also a constant time delayaffecting the density dependent growth rate of the prey. I fix𝜏1 at some value from its stability range (0, 𝜏̃10 ) and regard 𝜏2as a free parameter. I also assume that the model parametersare such, that hypothesis (H1 ) holds. Let 𝑖𝜔 with 𝜔 0 be aroot of (7). Then,𝜔4 𝑎̃𝜔2 2̃𝑏 sin (𝜔𝜏1 ) 2̃𝑐 cos (𝜔𝜏1 ) 𝑑̃ 0,(16)(20)(22)where𝑎̃ 𝐴2 𝐶12 𝐶22 ,̃𝑏 𝜔𝐶 𝐸 𝜔3 𝐶 ,21𝑐̃ 𝐷2 𝐸 𝜔2 𝐷1 𝜔2 𝐴𝐶1 ,𝑑̃ (𝐷22 𝐸2 ) .(23)

4International Journal of Ecology10060x908070PopulationE 504060y50y igure 1: Time series (a) and phase-plane (b) of the system (2). This figure shows that the coexistence equilibrium is asymptotically stable.Note that 𝑑̃ is always negative. I definẽ𝑓 (𝜔) 𝜔4 𝑎̃𝜔2 2̃𝑏 sin (𝜔𝜏1 ) 2̃𝑐 cos (𝜔𝜏1 ) 𝑑.the following result considering the change of stability ofsystem (3).(24)Then it is easy to verify that 𝑓(0) 0 and 𝑓(𝜔) , as𝜔 . Thus, (22) has finitely many positive roots, denoted̃2 , . . . , 𝜔̃𝑘 . For every fixed 𝜔̃𝑠 , where 𝑠 1, 2, . . . , 𝑘,̃1 , 𝜔by 𝜔there exists a sequence {̃𝜏2𝑡 𝑠 𝑡 N}, where𝜏̃2𝑡 𝑠 𝑀212𝑠𝜋cos 1 ( 𝑠 ) ,̃𝑠̃𝑠𝑁2𝑠𝜔𝜔𝑠 1, 2, . . . , 𝑘; 𝑡 N,(25)with̃𝑠 𝜏1 𝑅2 𝑆2 sin 𝜔̃𝑠 𝜏1 ,𝑀2𝑠 𝑃1 𝑆2 𝑃2 𝑇2 𝑅2 𝑇2 cos 𝜔𝑁2𝑠 𝑆22 𝑇22 ,̃𝑠2 ,𝑃1 𝜔̃𝑠 ,𝑃2 𝐴𝜔̃𝑠 ,𝑅2 𝐶1 𝜔(26)̃𝑠 𝜏1 𝐷2 ) ,𝑆2 (𝐸 cos 𝜔̃𝑠 𝜏1 𝐶2 𝜔𝑠 ,𝑇2 𝐸 sin 𝜔Case 4 (𝜏1 0, 𝜏2 ̸ 0). For this choice of the delay parameters, I summarize my results in the following theorem. Theproof follows similar arguments as the stability Theorem 2. inCase 2 above.Theorem 4. Assume that 𝜏1 0, 𝜏2 ̸ 0 and the conditions inhypothesis (𝐻1 ) hold true. Then the equilibrium 𝐸 (𝑥 , 𝑦 ) islocally asymptotically stable for 𝜏2 𝜏20 and unstable for 𝜏2 𝜏20 . Furthermore, the system (3) undergoes Hopf bifurcationwhen 𝜏2 𝜏20 , where𝜏20 𝑠 1, 2, . . . , 𝑘,such that (22) holds. Let 𝜏̃20 min{𝜏2𝑡 𝑠 𝑠 1, 2, . . . , 𝑘; 𝑡 1, 2, . . .}. When 𝜏2 𝜏̃20 , (7) has a pair of pure imaginary roots̃1 for 𝜏2 [0, 𝜏̃20 ). In the following, I assume that 𝑖𝜔(H5 )𝑑[(Re 𝜆(𝜏2 ))] ̸ 0.𝑑𝜏2̃0𝜆 𝑖𝜔Theorem 3. Suppose that the parameters in system (3) aresuch that hypotheses (𝐻1 ) and (𝐻3 ) hold true, and 𝜏1 [0, 𝜏̃10 ). Then the coexistence equilibrium 𝐸 (𝑥 , 𝑦 ) is locallyasymptotically stable when 𝜏2 (0, 𝜏̃20 ), and it is unstable when𝜏2 𝜏̃20 . Moreover, Hopf bifurcation occurs when 𝜏2 𝜏̃20 .(27)Therefore, by the general Hopf bifurcation theorem of functional differential equations, see, for example, [53], I obtain(𝐷 𝐸) 𝜔02 (𝐴 𝐶1 ) 𝐶2 𝜔021),cos 1 ( 22𝜔0𝐶2 𝜔2 (𝐷2 𝐸)(28)2 0and 𝜔0 is the unique positive root of the polynomial as follows:22𝜔4 [𝐶22 (𝐴 𝐶1 ) ] 𝜔2 (𝐷2 𝐸) 0.(29)Case 5 (𝜏1 ̸ 0, 𝜏2 ̸ 0 and 𝜏2 is within its stability range). Thischoice of parameter regime corresponds to the biologicallyinteresting case, when the predator individuals have a fixedgestation period and the growth of the prey populationis affected by delayed density dependent mechanisms. Theproof of the main result follows similar lines to that in Case

International Journal of Ecology5200150150y100y ��19896(a) 𝜏2 00.30.20.44440.80.70.6𝜏2(a) 𝜏1 02550Time(b) 𝜏1 0.45(b) 𝜏2 2550Time(c) 𝜏1 0.55(c) 𝜏2 0.48Figure 2: Bifurcation diagrams of the system (3) with respect to 𝜏1 inCase 2 in the three-dimensional space (𝜏1 , 𝑥, 𝑦). (b) is the time seriesfor (𝜏1 𝜏̃10 ) and (c) is the same for (𝜏1 𝜏̃10 ). These figures showthat the coexistence equilibrium is stable for 𝜏1 0.5329, unstablefor 𝜏1 0.5329, and a Hopf bifurcation exists at 𝜏1 0.5329.Figure 3: Bifurcation diagram of the system (3) with respect to 𝜏2in Case 3. (a) is drawn in the three-dimensional space (𝜏2 , 𝑥, 𝑦).(b) and (c) are the time evolutions for 𝜏2 0.4 𝜏̃20 0.4444(system is stable) and 𝜏2 0.48 𝜏̃20 0.4444 (system is unstable),respectively. A Hopf bifurcation exists at 𝜏2 0.4444. Here, 𝜏1 0.45 [0, 0.5329).

6International Journal of Ecology300250200100xy me𝜏20.2(a) 𝜏1 050(b) 𝜏2 0.4Population100x50y0025Time50(c) 𝜏2 0.47Figure 4: Bifurcation diagrams of the system (3) with respect to 𝜏2 in Case 4 in the three-dimensional space (𝜏2 , 𝑥, 𝑦). (b) is the time seriesfor (𝜏2 𝜏20 ) and (c) is the same for (𝜏2 𝜏20 ). These figures show that the coexistence equilibrium is stable for 𝜏2 0.4205, unstable for𝜏2 0.4205, and a Hopf bifurcation exists at 𝜏2 0.4205.3 and I only summarize the stability results in the followingtheorem.Theorem 5. Assume that the model parameters are such thatthe hypotheses in (𝐻1 ) hold true, and 𝜏2 [0, 𝜏20 ). Thenthe coexistence equilibrium 𝐸 (𝑥 , 𝑦 ) is asymptotically stablewhen 𝜏1 [0, 𝜏10 ), and it is unstable when 𝜏1 𝜏10 . Hopf𝑗bifurcation occurs when 𝜏1 𝜏10 , where 𝜏10 min{𝜏1𝑖 𝑖 1, 2, . . . , 𝑘; 𝑗 N} and𝑗𝜏1𝑖 (𝑀112𝑖𝜋) cos 1 ( 𝑖 ) ,𝜔𝑖𝑁1𝑖𝜔𝑖𝑖 1, 2, . . . , 𝑘; 𝑗 N,(30)with𝑀1𝑖 𝑃1 𝑆1 𝑃2 𝑇1 (𝑄1 𝑆1 𝑅1 𝑇1 ) cos 𝜔𝑖 𝜏2 (𝑅1 𝑆1 𝑄1 𝑇1 ) sin 𝜔𝑖 𝜏2 ,𝑁1𝑖 𝑆12 𝑇12 ,𝑄1 𝐷2 ,𝑃1 𝜔𝑖2 ,𝑃2 𝐴𝜔𝑖 ,𝑅1 𝐶2 𝜔𝑖 ,𝑆1 𝐸 cos 𝜔𝑖 𝜏2 ,𝑇1 𝐸 sin (𝜔𝑖 𝜏2 ) 𝐶1 𝜔𝑖 ,𝑖 1, 2, . . . , 𝑘.(31)3. Numerical ComputationsI perform numerical computations to observe variousdynamics of the coexistence equilibrium for both the systems(2) and (3). I consider the fixed parameter values as 𝑟 3, 𝑘 100, 𝛼 1.0769, 𝑎 10, 𝑠 3.5, ℎ 2, and 𝑚 0.35.

International Journal of Ecology7200200150Populationy 10050x100y0300200x10000.20.40.300.5296 0.6025Time𝜏1(a) 𝜏2 0.450(b) 𝜏1 0.48200Populationx1000y025Time50(c) 𝜏1 0.54Figure 5: Bifurcation diagram of the system (3) with respect to 𝜏1 in Case 5. (a) is drawn in the three-dimensional space (𝜏1 , 𝑥, 𝑦). (b) and(c) are the time evolutions for 𝜏1 0.48 𝜏10 0.5296 (system is stable) and 𝜏1 0.54 𝜏10 0.5296 (system is unstable), respectively. AHopf bifurcation exists at 𝜏1 0.5296. Here, 𝜏2 0.4 [0, 0.4205).With these parameter values 𝑥 98.0556, 𝑦 49.0278.Here, 𝐴 𝐶1 𝐶2 6.4319 0 and 𝐷2 𝐸 10.2618 0,and initial value is considered as (12, 5) for each simulation.First I consider the nondelayed system (2). In this case, thesystem is asymptotically stable (Figure 1).I now study the dynamic behavior of the delay-inducedsystem (3). When 𝜏1 ̸ 0, 𝜏2 0 (Case 2), one can computẽ0 2.9417, 𝜏10 0.5329, and (𝐴 from Theorem 2 that 𝜔𝐶2)2 𝐶12 2𝐷2 3.5967 0 & 𝐷22 𝐸2 106.003 0.Therefore, the coexisting equilibrium 𝐸 (𝑥 , 𝑦 ) is asymptotically stable for 𝜏1 0.45 𝜏̃10 0.5329 (Figure 2(b))and unstable for 𝜏1 0.55 𝜏̃10 0.5329 (Figure 2(c)).When 𝜏1 𝜏̃10 , the system (3) undergoes a Hopf bifurcation at𝐸 (𝑥 , 𝑦 ). System behaviors in Case 2 can be demonstratedmore prominently if I plot the bifurcation diagram in thethree-dimensional space (𝜏1 , 𝑥, 𝑦). Figure 2(a) shows that,when 𝜏2 0, the coexistence equilibrium is stable for 𝜏1 𝜏̃10 0.5329 but the instability sets in when 𝜏1 𝜏̃10 0.5329.For Case 3, I take any value of 𝜏1 from its stabilityrange [0, 0.5329), say that 𝜏1 0.45, and consider 𝜏2 as̃1 as 3.5345a parameter. One can compute the value of 𝜔and the corresponding critical value of 𝜏2 as 𝜏̃20 0.4444.Thus, for fixed stable value of 𝜏1 ( 0.45), the system (3)exhibits stable behavior around 𝐸 (𝑥 , 𝑦 ) for 𝜏2 𝜏̃20 andunstable oscillatory behavior for 𝜏2 𝜏̃20 (Figure 3). A Hopfbifurcation occurs when 𝜏2 takes the critical value 𝜏̃20 0.4444.In Case 4, when 𝜏1 0, 𝜏2 ̸ 0, one can compute fromTheorem 4, 𝜔0 3.7359 and 𝜏20 0.4205. One can alsoverify that (𝐴 𝐶1 )2 𝐶2 2 3.6537 0 and (𝐷2 𝐸)2 105.3047 0. Therefore, the coexisting equilibrium𝐸 (𝑥 , 𝑦 ) is asymptotically stable for 𝜏2 𝜏20 , unstable for

8International Journal of Ecology200250200150150x igure 6: Time evolution (a)-(b) and phase-plane diagram (c) of the system (3) for 𝜏1 0.56 and 𝜏2 0.535. Here, the system (3) exhibitschaotic behavior. Other parameters are as in previous figures.𝜏2 𝜏20 , and when 𝜏2 𝜏20 , the system (3) undergoes a Hopfbifurcation at 𝐸 (𝑥 , 𝑦 ). These results can be observed fromthe bifurcation diagram Figure 4.In the last case (Case 5), I take any value of 𝜏2 from itsstability range [0, 0.4205), say that 𝜏2 0.4, and consider𝜏1 as a parameter. As before, I calculate 𝜔1 2.96 and thecorresponding critical value 𝜏10 0.5296. The bifurcationdiagram Figure 5 demonstrates that the system (2) is locallyasymptotically stable around 𝐸 (𝑥 , 𝑦 ) when 𝜏1 0.5296and unstable when 𝜏1 0.5296. These results can be observedfrom the bifurcation diagram Figure 5.3.1. Chaotic Dynamics. I have analytically studied the localbehavior of the system (3) when the delay parameters arewithin or slightly above their critical values. One interestingtopic in the delay-induced system is to study the dynamicalbehavior of the system when the delay parameters are faraway from their critical values, or they assume large values.To observe the system dynamics, I consider 𝜏1 0.56 and𝜏2 0.535, both are beyond their stability range, and thenthe system shows chaotic behavior (Figure 6).In order to characterize the irregular behavior, I performthe standard numerical diagnostics, for example, Lyapunovexponent, sensitivity of the solutions to initial condition,power spectral density, and recurrence plot. Figure 7(a)shows the Lyapunov exponent of 𝑦 with respect to the delay 𝜏2in a smaller range [0.45, 0.539] for fixed 𝜏1 . Positive Lyapunovexponent (𝜆) indicates that the system exhibits chaoticbehavior [42, 54, 55]. Sensitivity of the system trajectoriesto the initial values is shown in Figure 7(b). To show this,I compute the error Δ𝑦(𝑡) 𝑦1 (𝑡) 𝑦2 (𝑡), where 𝑦1 (0) (12, 5) and 𝑦2 (0) (12.001, 5). This phenomena ensures thechaotic nature of the system [42, 54, 56]. The power spectraof the predator population are presented in Figure 7(c). Theirregular broad peaks of this figure are indicative of chaosand randomness [42, 54, 57]. The recurrence plot of thesystem for the parameter values as in Figure 6 is representedin Figure 7(d). The random points on the time-time plane

International Journal of Ecology9400Δy(t) y1 (t) y2 (t)Lyapunov exponent (𝜆)1.510.500.450.49452000 200 mePower/frequency (dB/Hz)10000400200 50000.20.40.6Frequency (Hz)0.81(c)02004006008001000Time(d)Figure 7: Standard diagnostic tests to detect chaos. (a) Lyapunov exponent, (b) sensitivity on the initial condition, (c) power spectral density,and (d) recurrence plot. Parameters are as in Figure 6.ensure that more or less the same values of phase trajectoriestake place without any rhythm, indicating the chaotic natureof the system [42, 54, 58].3.2. Biological Control of Chaotic Dynamics. Many ecologicalsituations show that their unstable or abnormal behavior maybe controlled by any external perturbation [14, 21] or habitatstructure [34, 36–40]. Recently Jana [42] shows that chaoticbehavior of a discrete predator-prey system at low level ofprey refuge must be controlled to its stable coexistence byincreasing the degree of prey refuge. Parameter set of Figure 6depicts the chaotic behavior of the system (3). Standing at thissituation, if I increase the intensity of degree of prey refuge(from 0.35 to 0.6), then again the system (3) settles down toits stable coexistence (Figure 8).4. DiscussionEffect of prey refuge has an extended impact on the predatorprey models. However, both field and laboratory experimentsconfirm that intensity of prey refuge reduces predation ratesby decreasing encounter rates between predator and prey.On the other hand, a predator-prey model becomes morerealistic in presence of different delays which are unavoidableelements in physiological and ecological processes. In thispaper, I have studied a multidelayed predator-prey modelwhere the prey species is subject to partial refuge frompredator population. Also I have considered here that thepredator is a generalist type. A time delay is considered inthe growth rate of the prey to represent density dependentfeedback mechanism, and a second delay is introduced toaccount for the gestation time of the predator. The objectivewas to study the role of the effects of the interaction ofmultiple constant time delays and the prey refuge on thedynamical behavior of a predator-prey system. To have abetter understanding of the complexities in natural systems,I can construct larger systems of differential equations containing more parameters. A second approach that is gainingprominence is the inclusion of time delay terms in the differential equations. Complex dynamical behaviour arises as aconsequence of time delay in a biological system (with significant time delay) which may exhibit limited cycle oscillationand chaos. In case of the multidelayed system, I obtainedsufficient conditions in terms of the system parameters for

10International Journal of Ecology10050yPopulation90804070356030x50y 2540203015201010500246E 458100020406080100xTime(a)(b)Figure 8: Biological control of chaos with increasing refuge 𝑚 0.6. Other parameters are as in Figure 6.the stability of the coexistence equilibrium. I observed thatthe coexistence equilibrium is locally asymptotically stable ifthe delay parameters are lower than some critical values andit is unstable when the parameter values exceed these criticalvalues. One important question in the multidelayed predatorprey system is to study the behavior of the system when thedelay parameter is significantly large and hence it is far awayfrom its critical value. As the estimated length of delays topreserve stability and the critical length of time delays forHopf bifurcation are dependent on the system’s parameters,it is possible to impose some control, which will prevent thepossible abnormal oscillation in the population density. Toexplore the behavior of the system in this case, I performedextensive numerical simulations. My simulation results showthat a refuged predator-prey system with multiple delaysmay exhibit different interesting (e.g., chaotic) behavior whenthe delay is large enough. Finally I showed that the chaoticcharacteristic of the multidelayed system was controlled byincreasing the intensity of the prey refuge.Conflict of InterestsThe author declares that there is no conflict of interestsregarding the publication of this paper.AcknowledgmentThis research is supported by UGC (Dr. D. S. KothariPostdoctoral Fellowship), India; no. F.4-2/2006(BSR)/131004/2013(BSR).References[1] J. Matthiopoulos, K. Graham, S. Smout et al., “Sensitivity toassumptions in models of generalist predation on a cyclic prey,”Ecology, vol. 88, no. 10, pp. 2576–2586, 2007.[2] T. M. Temesgen, “Bifurcation analysis on the dynamics of a genralist predator-prey system,” International Journal of Ecosystem,vol. 2, no. 3, pp. 38–43, 2012.[3] Y. Kang and L. Wedekin, “Dynamics of a intraguild predationmodel with generalist or specialist predator,” Journal of Mathematical Biology, vol. 67, no. 5, pp. 1227–1259, 2013.[4] R. K. Upadhyay and S. R. K. Iyengar, Introduction to Mathematical Modelling and Chaotic Dynamics, Taylor and Francis, 2013.[5] Q. Wang, M. Fan, and K. Wang, “Dynamics of a class of nonautonomous semi-ratio-dependent predator-prey systems withfunctional responses,” Journal of Mathematical Analysis andApplications, vol. 278, no. 2, pp. 443–471, 2003.[6] X.-X. Liu, “A note on periodic solutions for semi-ratio-dependent predator-prey systems,” Applied Mathematics, vol. 25, no.1, pp. 1–8, 2010.[7] L. Hongying and W. Weiguo, “Dynamics of a delayed discretesemiratiodependent predator-prey system with Holling type IVfunctional response,” Advances in Difference Equation, vol. 7, pp.2–19, 2011.[8] C. Lu and L. Zhang, “Permanence and global attractivity ofa discrete semi-ratio dependent predator-prey system withHolling II type functional response,” Journal of Applied Mathematics and Computing, vol. 33, no. 1-2, pp. 125–135, 2010.[9] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, NJ, USA, 1974.[10] J. D. Murray, Mathematical Biology, Springer, Berlin, Germany,1989.

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Stabilizing Effect of Prey Refuge and Predator s Interference on the Dynamics of Prey with Delayed Growth and Generalist Predator with Delayed Gestation DebaldevJana . hypothesis (-1) hold true. e n the equilibrium ! ( , ) is locally asymptotically stable for 2 2 0 and unstable for 2 2 0.Furthermore,thesystem( ) undergoes Hopf bifurcation