April  2021, 26(4): 1889-1915. doi: 10.3934/dcdsb.2020281

On predation effort allocation strategy over two patches

1. 

Department of Applied Mathematics, National Pingtung University, Pingtung 90003, Taiwan

2. 

Department of Applied Mathematics, University of Western Ontario, London, ON, Canada N6A 5B7

* Corresponding author: Xingfu Zou

Received  August 2020 Published  April 2021 Early access  September 2020

Fund Project: This paper is dedicated to Professor Sze-Bi Hsu in celebrating his retirement and in honour of his great contributions to the fields of mathematical biology and dynamical systems. C.-Y. Cheng was partially supported by the Ministry of Science and Technology, Taiwan, R.O.C. (Grant No. MOST 108-2115-M-153-003); X. Zou was partially supported by NSERC of Canada (RGPIN-2016-04665)

In this paper, we formulate an ODE model to describe the population dynamics of one non-dispersing prey and two dispersing predators in a two-patch environment with spatial heterogeneity. The dispersals of the predators are implicitly reflected by the allocation of their presence (foraging time) in each patch. We analyze the dynamics of the model and discuss some biological implications of the theoretical results on the dynamics of the model. Particularly, we relate the results to the evolution of the allocation strategy and explore the impact of the spatial heterogeneity and the difference in fitness of the two predators on the allocation strategy. Under certain range of other parameters, we observe the existence of an evolutionarily stable strategy (ESS) while in some other ranges, the ESS disappears. We also discuss some possible extensions of the model. Particularly, when the model is modified to allow distinct preys in the two patches, we find that the heterogeneity in predation rates and biomass transfer rates in the two patches caused by such a modification may lead to otherwise impossible bi-stability for some pairs of equilibria.

Citation: Chang-Yuan Cheng, Xingfu Zou. On predation effort allocation strategy over two patches. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 1889-1915. doi: 10.3934/dcdsb.2020281
References:
[1]

R. S. CantrellC. CosnerD. L. Deangelis and V. Padron, The ideal free distribution as an evolutionarily stable strategy, J. Biol. Dyn., 1 (2007), 249-271.  doi: 10.1080/17513750701450227.

[2]

R. S. CantrellC. Cosner and Y. Lou, Evolutionary stability of ideal free dispersal strategies in patchy environments, J. Math. Biol., 65 (2012), 943-965.  doi: 10.1007/s00285-011-0486-5.

[3]

R. S. CantrellC. Cosner and S. Ruan, Intraspecific interference and consumer-resource dynamics, Disc. Cont. Dyn. Syst. B, 4 (2004), 527-546.  doi: 10.3934/dcdsb.2004.4.527.

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J. Colbert, E. Danchin, A. A. Dhondt and J. D. Nichols, Dispersal, Oxford University Press, New York, 2001.

[5] S. Creel and N. M. Dreel, The African Wild Dog: Behavior, Ecology and Conservation, Princeton University Press, Princeton, 2002. 
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R. CressmanV. Křivan and J. Garay, Ideal free distributions, evolutionary games, and population dynamics in multiple-species environments, Amer. Natur., 164 (2004), 473-489.  doi: 10.1086/423827.

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R. Cressman and V. Křivan, Migration dynamics for the ideal free distribution, Amer. Natur., 168 (2006), 384-397.  doi: 10.1086/506970.

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J. CuiY. Takeuchi and Z. Lin, Permanence and extinction for dispersal population systems, J. Math. Anal. Appl., 298 (2004), 73-93.  doi: 10.1016/j.jmaa.2004.02.059.

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J. DockeryV. HutsonK. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction-diffusion model, J. Math. Biol., 37 (1998), 61-83.  doi: 10.1007/s002850050120.

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S. D. Fretwell and H. L. Lucas, On territorial behavior and other factors influencing habitat selection in birds, Acta Biotheor., 19 (1969), 16-36.  doi: 10.1007/BF01601953.

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J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395.  doi: 10.1137/0520025.

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X. He and W.-M. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition diffusion system I: Heterogeneity vs. homogeneity, J. Differ. Eqs., 254 (2013), 528-546.  doi: 10.1016/j.jde.2012.08.032.

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S. B. HsuS. P. Hubbell and P. Waltman, Competing predators, SIAM J. Appl. Math., 35 (1978), 617-625.  doi: 10.1137/0135051.

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S. B. HsuS. P. Hubbell and P. Waltman, A contribution to the theory of competing predators, Ecological Monographs, 48 (1978), 337-349.  doi: 10.2307/2937235.

[15]

Y. KangS. K. Sasmal and K. Messan, Two-patch prey-predator model with predator dispersal driven by the predation strength, Math. Biosci. Eng., 14 (2017), 843-880.  doi: 10.3934/mbe.2017046.

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V. Křvan, The Lotka-Volterra predator-prey model with foraging-predation risk trade-offs, Amer. Natur., 170 (2007), 771-782.  doi: 10.1086/522055.

[17]

Y. Kuang and Y. Takeuchi, Predator-prey dynamics in models of prey dispersal in two-patch environments, Math. Biosci., 120 (1994), 77-98.  doi: 10.1016/0025-5564(94)90038-8.

[18]

M. KummelD. Brown and A. Bruder, How the aphids got their spots: Predation drives self-organization of aphid colonies in a patchy habitat, Oikos, 122 (2013), 896-906.  doi: 10.1111/j.1600-0706.2012.20805.x.

[19]

S. A. Levin, Community equilibria and stability, and an extension of the competitive exclusion principle, Amer. Natur., 104 (1970), 413-423.  doi: 10.1086/282676.

[20]

J. Llibre and D. Xiao, Global dynamics of a Lotka-Volterra model with two predators competing for one prey, SIAM J. Appl. Math., 74 (2014), 434-453.  doi: 10.1137/130923907.

[21]

Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differ. Eqs., 223 (2006), 400-426.  doi: 10.1016/j.jde.2005.05.010.

[22]

S. Lowe, M. Browne and S. Boudjelas, 100 of the World's Worst Invasive Alien Species. A Selection from the Global Invasive Species Database, Invasive Species Specialist Group, Auckland, New Zealand, 2000.

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R. H. MacArthur, Geographical Ecology, Harper and Row, New York, USA, 1972.

[24]

R. MacArthur and R. Levins, Competition, habitat selection, and character displacement in a patchy environment, Proc. Natl. Acad. Sci. USA, 51 (1964), 1207-1210.  doi: 10.1073/pnas.51.6.1207.

[25]

M. Martcheva and S. S. Pilyugin, The role of coinfection in multi-disease dynamics, SIAM J. Appl. Math., 66 (2006), 843-872.  doi: 10.1137/040619272.

[26]

R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, NJ, 1973.

[27]

K. Messan and Y. Kang, A two patch prey-predator model with multiple foraging strategies in predators, Disc. Cont. Dyn. Syst. B, 22 (2017), 947-976.  doi: 10.3934/dcdsb.2017048.

[28]

D. W. Morris, Adaptation and habitat selection in the eco-evolutionary process, Proc. R. Soc. Lond. B, 27 (2011), 2401-2411.  doi: 10.1098/rspb.2011.0604.

[29]

J. PassargeS. HolM. Escher and J. Huisman, Competition for nutrients and light: Stable coexistence, alternative stable states, or competitive exclusion?, Ecological Monographs, 76 (2006), 57-72.  doi: 10.1890/04-1824.

[30]

D. Pimentel, Biological Invasions: Economic and Environmental Costs of Alien Plant, Animal, and Microbe Species, CRC Press, New York, 2002. doi: 10.1201/9781420041668.

[31]

H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407-435.  doi: 10.1137/0524026.

[32]

X. Wang and X. Zou, On a two-patch predator-prey model with adaptive habitancy of predators, Disc. Cont. Dyn. Syst. B, 21 (2016), 677-697.  doi: 10.3934/dcdsb.2016.21.677.

[33]

D. K. Wasko and M. Sasa, Food resources influence spatial ecology, habitat selection, and foraging behavior in an ambush-hunting snake (Viperidae: Bothrops asper): An experimental study, Zoology, 115 (2012), 179-187.  doi: 10.1016/j.zool.2011.10.001.

[34]

M. H. Williamson, Biological Invasions, Chapman and Hall, London, 1996.

[35]

X.-Q. Zhao, Uniform persistence and periodic coexistence states in infinite dimensional periodic semiflows with applications, Can. Appl. Math. Quart.B, 3 (1995), 473-495. 

show all references

References:
[1]

R. S. CantrellC. CosnerD. L. Deangelis and V. Padron, The ideal free distribution as an evolutionarily stable strategy, J. Biol. Dyn., 1 (2007), 249-271.  doi: 10.1080/17513750701450227.

[2]

R. S. CantrellC. Cosner and Y. Lou, Evolutionary stability of ideal free dispersal strategies in patchy environments, J. Math. Biol., 65 (2012), 943-965.  doi: 10.1007/s00285-011-0486-5.

[3]

R. S. CantrellC. Cosner and S. Ruan, Intraspecific interference and consumer-resource dynamics, Disc. Cont. Dyn. Syst. B, 4 (2004), 527-546.  doi: 10.3934/dcdsb.2004.4.527.

[4]

J. Colbert, E. Danchin, A. A. Dhondt and J. D. Nichols, Dispersal, Oxford University Press, New York, 2001.

[5] S. Creel and N. M. Dreel, The African Wild Dog: Behavior, Ecology and Conservation, Princeton University Press, Princeton, 2002. 
[6]

R. CressmanV. Křivan and J. Garay, Ideal free distributions, evolutionary games, and population dynamics in multiple-species environments, Amer. Natur., 164 (2004), 473-489.  doi: 10.1086/423827.

[7]

R. Cressman and V. Křivan, Migration dynamics for the ideal free distribution, Amer. Natur., 168 (2006), 384-397.  doi: 10.1086/506970.

[8]

J. CuiY. Takeuchi and Z. Lin, Permanence and extinction for dispersal population systems, J. Math. Anal. Appl., 298 (2004), 73-93.  doi: 10.1016/j.jmaa.2004.02.059.

[9]

J. DockeryV. HutsonK. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction-diffusion model, J. Math. Biol., 37 (1998), 61-83.  doi: 10.1007/s002850050120.

[10]

S. D. Fretwell and H. L. Lucas, On territorial behavior and other factors influencing habitat selection in birds, Acta Biotheor., 19 (1969), 16-36.  doi: 10.1007/BF01601953.

[11]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395.  doi: 10.1137/0520025.

[12]

X. He and W.-M. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition diffusion system I: Heterogeneity vs. homogeneity, J. Differ. Eqs., 254 (2013), 528-546.  doi: 10.1016/j.jde.2012.08.032.

[13]

S. B. HsuS. P. Hubbell and P. Waltman, Competing predators, SIAM J. Appl. Math., 35 (1978), 617-625.  doi: 10.1137/0135051.

[14]

S. B. HsuS. P. Hubbell and P. Waltman, A contribution to the theory of competing predators, Ecological Monographs, 48 (1978), 337-349.  doi: 10.2307/2937235.

[15]

Y. KangS. K. Sasmal and K. Messan, Two-patch prey-predator model with predator dispersal driven by the predation strength, Math. Biosci. Eng., 14 (2017), 843-880.  doi: 10.3934/mbe.2017046.

[16]

V. Křvan, The Lotka-Volterra predator-prey model with foraging-predation risk trade-offs, Amer. Natur., 170 (2007), 771-782.  doi: 10.1086/522055.

[17]

Y. Kuang and Y. Takeuchi, Predator-prey dynamics in models of prey dispersal in two-patch environments, Math. Biosci., 120 (1994), 77-98.  doi: 10.1016/0025-5564(94)90038-8.

[18]

M. KummelD. Brown and A. Bruder, How the aphids got their spots: Predation drives self-organization of aphid colonies in a patchy habitat, Oikos, 122 (2013), 896-906.  doi: 10.1111/j.1600-0706.2012.20805.x.

[19]

S. A. Levin, Community equilibria and stability, and an extension of the competitive exclusion principle, Amer. Natur., 104 (1970), 413-423.  doi: 10.1086/282676.

[20]

J. Llibre and D. Xiao, Global dynamics of a Lotka-Volterra model with two predators competing for one prey, SIAM J. Appl. Math., 74 (2014), 434-453.  doi: 10.1137/130923907.

[21]

Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differ. Eqs., 223 (2006), 400-426.  doi: 10.1016/j.jde.2005.05.010.

[22]

S. Lowe, M. Browne and S. Boudjelas, 100 of the World's Worst Invasive Alien Species. A Selection from the Global Invasive Species Database, Invasive Species Specialist Group, Auckland, New Zealand, 2000.

[23]

R. H. MacArthur, Geographical Ecology, Harper and Row, New York, USA, 1972.

[24]

R. MacArthur and R. Levins, Competition, habitat selection, and character displacement in a patchy environment, Proc. Natl. Acad. Sci. USA, 51 (1964), 1207-1210.  doi: 10.1073/pnas.51.6.1207.

[25]

M. Martcheva and S. S. Pilyugin, The role of coinfection in multi-disease dynamics, SIAM J. Appl. Math., 66 (2006), 843-872.  doi: 10.1137/040619272.

[26]

R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, NJ, 1973.

[27]

K. Messan and Y. Kang, A two patch prey-predator model with multiple foraging strategies in predators, Disc. Cont. Dyn. Syst. B, 22 (2017), 947-976.  doi: 10.3934/dcdsb.2017048.

[28]

D. W. Morris, Adaptation and habitat selection in the eco-evolutionary process, Proc. R. Soc. Lond. B, 27 (2011), 2401-2411.  doi: 10.1098/rspb.2011.0604.

[29]

J. PassargeS. HolM. Escher and J. Huisman, Competition for nutrients and light: Stable coexistence, alternative stable states, or competitive exclusion?, Ecological Monographs, 76 (2006), 57-72.  doi: 10.1890/04-1824.

[30]

D. Pimentel, Biological Invasions: Economic and Environmental Costs of Alien Plant, Animal, and Microbe Species, CRC Press, New York, 2002. doi: 10.1201/9781420041668.

[31]

H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407-435.  doi: 10.1137/0524026.

[32]

X. Wang and X. Zou, On a two-patch predator-prey model with adaptive habitancy of predators, Disc. Cont. Dyn. Syst. B, 21 (2016), 677-697.  doi: 10.3934/dcdsb.2016.21.677.

[33]

D. K. Wasko and M. Sasa, Food resources influence spatial ecology, habitat selection, and foraging behavior in an ambush-hunting snake (Viperidae: Bothrops asper): An experimental study, Zoology, 115 (2012), 179-187.  doi: 10.1016/j.zool.2011.10.001.

[34]

M. H. Williamson, Biological Invasions, Chapman and Hall, London, 1996.

[35]

X.-Q. Zhao, Uniform persistence and periodic coexistence states in infinite dimensional periodic semiflows with applications, Can. Appl. Math. Quart.B, 3 (1995), 473-495. 

Figure 1.  Different regions of $ \mu $ to examine the criterion $ F(\alpha_1, \alpha_2)>0 $ (when $ \xi>1 $)
Figure 2.  Strategies $ \alpha_1 $ vs $ \alpha_2 $ for $ F(\alpha_1, \alpha_2)>0 $ (with $ \xi>1 $): (A) for $ \mu $ in I-1, (B) for $ \mu $ in I-2 and I-3, (C) for $ \mu $ in I-4
Figure 3.  Strategies $ \alpha_1 $ vs $ \alpha_2 $ for $ G(\alpha_1, \alpha_2)>0 $ (with $ \xi>1 $). (A) for $ \mu $ in the region I-1, (B) for $ \mu $ in the regions I-2 and I-3, (C) for $ \mu $ in the region I-4
Figure 4.  $ \tilde{G}(\alpha_1, \alpha_2)>0 $ for $ p = 0.5 $, $ e = 0.2 $, $ K_1 = 1 $, $ \xi = 2 $ ($ K_2 = 2 $), $ \mu_1 = 0.022 $ and $ \mu_2 = 0.02 $
Figure 5.  $ \tilde{G}(\alpha_1, \alpha_2)>0 $ for $ p = 0.5 $, $ e = 0.2 $, $ K_1 = 1 $, $ \xi = 2 $ ($ K_2 = 2 $), $ \mu_1 = 0.018 $ and $ \mu_2 = 0.02 $
Figure 6.  Bi-stability in the system (1) with $ r = 1 $, $ K_1 = K_2 = 1 $, $ p_1 = p_2 = 0.5 $, $ \mu_1 = \mu_2 = 0.1333 $, $ e_{11} = 0.2 $, $ e_{12} = 0.05 $, $ e_{21} = 0.05 $ and $ e_{22} = 0.2 $, $ \alpha_1 = \frac{1}{3} $ and $ \alpha_2 = \frac{2}{3} $. A solution either converges to $ E^1_1 $ or $ E^2_2 $ depending on the initial condition
Figure 7.  Bi-stability in the system (1) with $ r = 1 $, $ K_1 = K_2 = 1 $, $ p = 0.5 $, $ \mu = 0.025 $, $ e_{11} = 0.5 $, $ e_{12} = 0.1 $, $ e_{21} = 0.3 $ and $ e_{22} = 0.3 $, $ \alpha_1 = 0.4 $ and $ \alpha_2 = 0.6 $. A solution either converges to $ E^1_1 $ or $ E^2_3 $ depending on the initial condition
Figure 8.  Bi-stability in the system (1) with $ r = 1 $, $ K_1 = K_2 = 1 $, $ p = 0.5 $, $ \mu = 0.05 $, $ e_{11} = 0.2 $, $ e_{12} = 0.1 $, $ e_{21} = 0.1 $ and $ e_{22} = 0.2 $, $ \alpha_1 = 0.4 $ and $ \alpha_2 = 0.6 $. A solution either converges to $ E^1_3 $ or $ E^2_3 $ depending on the initial condition
Table 1.  Existence and stability of boundary equilibria for system (1). Cond-1: $ \frac{\mu_2(\mathcal{R}^2_0-1)}{p_2e_2(\alpha_1\alpha_2K_1+\beta_1\beta_2K_2)}<\frac{\mu_1(\mathcal{R}^1_0-1)}{p_1e_1(\alpha_1^2K_1+\beta_1^2K_2)} $; Cond-2: $ \frac{\mu_1(\mathcal{R}^1_0-1)}{p_1e_1(\alpha_1\alpha_2K_1+\beta_1\beta_2K_2)}<\frac{\mu_2(\mathcal{R}^2_0-1)}{p_2e_2(\alpha_2^2K_1+\beta_2^2K_2)} $; LS: locally stable
Equilibrium Existence Stability Condition for stability
$ E^0_0, E^0_1, E^0_2 $ always exists unstable
$ E^0_3 $ always exists LS $ \mathcal{R}^1_0<1, \mathcal{R}^2_0<1 $
$ E^1_1 $ $ \mathcal{R}^1_1>1 $ LS $ \hat{\mathcal{R}}^1_1>1 $, $ \mathcal{R}_1^2<\mathcal{R}_1^1 $
$ E^1_2 $ $ \mathcal{R}^1_2>1 $ LS $ \hat{\mathcal{R}}^1_2>1 $, $ \mathcal{R}_2^2<\mathcal{R}_2^1 $
$ E^1_3 $ $ \mathcal{R}^1_0>1 $, $ \hat{\mathcal{R}}^1_1<1 $, $ \hat{\mathcal{R}}^1_2<1 $ LS Cond-1
$ E^2_1 $ $ \mathcal{R}^2_1>1 $ LS $ \hat{\mathcal{R}}^2_1>1 $, $ \mathcal{R}_1^1<\mathcal{R}_1^2 $
$ E^2_2 $ $ \mathcal{R}^2_2>1 $ LS $ \hat{\mathcal{R}}^2_2>1 $, $ \mathcal{R}_2^1<\mathcal{R}_2^2 $
$ E^2_3 $ $ \mathcal{R}^2_0>1 $, $ \hat{\mathcal{R}}^2_1<1 $, $ \hat{\mathcal{R}}^2_2<1 $ LS Cond-2
$ E^* $ Theorem 3.8
Equilibrium Existence Stability Condition for stability
$ E^0_0, E^0_1, E^0_2 $ always exists unstable
$ E^0_3 $ always exists LS $ \mathcal{R}^1_0<1, \mathcal{R}^2_0<1 $
$ E^1_1 $ $ \mathcal{R}^1_1>1 $ LS $ \hat{\mathcal{R}}^1_1>1 $, $ \mathcal{R}_1^2<\mathcal{R}_1^1 $
$ E^1_2 $ $ \mathcal{R}^1_2>1 $ LS $ \hat{\mathcal{R}}^1_2>1 $, $ \mathcal{R}_2^2<\mathcal{R}_2^1 $
$ E^1_3 $ $ \mathcal{R}^1_0>1 $, $ \hat{\mathcal{R}}^1_1<1 $, $ \hat{\mathcal{R}}^1_2<1 $ LS Cond-1
$ E^2_1 $ $ \mathcal{R}^2_1>1 $ LS $ \hat{\mathcal{R}}^2_1>1 $, $ \mathcal{R}_1^1<\mathcal{R}_1^2 $
$ E^2_2 $ $ \mathcal{R}^2_2>1 $ LS $ \hat{\mathcal{R}}^2_2>1 $, $ \mathcal{R}_2^1<\mathcal{R}_2^2 $
$ E^2_3 $ $ \mathcal{R}^2_0>1 $, $ \hat{\mathcal{R}}^2_1<1 $, $ \hat{\mathcal{R}}^2_2<1 $ LS Cond-2
$ E^* $ Theorem 3.8
Table 2.  Conditions for $ F(\alpha_1, \alpha_2)>0 $ in variant values of $ \mu $
Value of $ \mu $ I-1 I-2 I-3 I-4
Conditions $ \left\{ \begin{array}{l} \alpha_1>\alpha_2\\ \alpha_2<b/a \end{array} \right. $ or $ \alpha_1<\alpha_2 $ $ \alpha_1<\alpha_2 $ $ \left\{ \begin{array}{l} \alpha_1>\alpha_2\\ \alpha_2>b/a \end{array} \right. $ or
$ \left\{ \begin{array}{l} \alpha_1<\alpha_2\\ \alpha_2>b/a \end{array} \right. $ $ \left\{ \begin{array}{l} \alpha_1<\alpha_2\\ \alpha_2<b/a \end{array} \right. $
Value of $ \mu $ I-1 I-2 I-3 I-4
Conditions $ \left\{ \begin{array}{l} \alpha_1>\alpha_2\\ \alpha_2<b/a \end{array} \right. $ or $ \alpha_1<\alpha_2 $ $ \alpha_1<\alpha_2 $ $ \left\{ \begin{array}{l} \alpha_1>\alpha_2\\ \alpha_2>b/a \end{array} \right. $ or
$ \left\{ \begin{array}{l} \alpha_1<\alpha_2\\ \alpha_2>b/a \end{array} \right. $ $ \left\{ \begin{array}{l} \alpha_1<\alpha_2\\ \alpha_2<b/a \end{array} \right. $
Table 3.  Conditions for $ G(\alpha_1, \alpha_2)>0 $ in variant values of $ \mu $
Value of $ \mu $ I-1 I-2 I-3 I-4
Conditions $ \left\{ \begin{array}{l} \alpha_1>\alpha_2\\ \alpha_1<b/a \end{array} \right. $ or $ \alpha_1<\alpha_2 $ $ \alpha_1<\alpha_2 $ $ \left\{ \begin{array}{l} \alpha_1>\alpha_2\\ \alpha_1>b/a \end{array} \right. $ or
$ \left\{ \begin{array}{l} \alpha_1<\alpha_2\\ \alpha_1>b/a \end{array} \right. $ $ \left\{ \begin{array}{l} \alpha_1<\alpha_2\\ \alpha_1<b/a \end{array} \right. $
Value of $ \mu $ I-1 I-2 I-3 I-4
Conditions $ \left\{ \begin{array}{l} \alpha_1>\alpha_2\\ \alpha_1<b/a \end{array} \right. $ or $ \alpha_1<\alpha_2 $ $ \alpha_1<\alpha_2 $ $ \left\{ \begin{array}{l} \alpha_1>\alpha_2\\ \alpha_1>b/a \end{array} \right. $ or
$ \left\{ \begin{array}{l} \alpha_1<\alpha_2\\ \alpha_1>b/a \end{array} \right. $ $ \left\{ \begin{array}{l} \alpha_1<\alpha_2\\ \alpha_1<b/a \end{array} \right. $
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