American Institute of Mathematical Sciences

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doi: 10.3934/dcdsb.2020281

On predation effort allocation strategy over two patches

Chang-Yuan Cheng 1, and  Xingfu Zou 2,,

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  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.

Keywords: Predator-prey, resource competition, spatial heterogeneity, population dynamics, evolutionarily stable strategy.
Mathematics Subject Classification: Primary: 92-10, 92D25; Secondary: 34D05, 34D23.
Citation: Chang-Yuan Cheng, Xingfu Zou. On predation effort allocation strategy over two patches. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020281
References:
[1]

R. S. Cantrell, C. Cosner, D. 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.  Google Scholar

[2]

R. S. Cantrell, C. 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.  Google Scholar

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R. S. Cantrell, C. 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.  Google Scholar

[4]

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

[5] S. Creel and N. M. Dreel, The African Wild Dog: Behavior, Ecology and Conservation, Princeton University Press, Princeton, 2002.   Google Scholar
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R. Cressman, V. 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.  Google Scholar

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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.  Google Scholar

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J. Cui, Y. 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.  Google Scholar

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J. Dockery, V. Hutson, K. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

[14]

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

[15]

Y. Kang, S. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[18]

M. Kummel, D. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[29]

J. Passarge, S. Hol, M. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[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.   Google Scholar

show all references

References:
[1]

R. S. Cantrell, C. Cosner, D. 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.  Google Scholar

[2]

R. S. Cantrell, C. 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.  Google Scholar

[3]

R. S. Cantrell, C. 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.  Google Scholar

[4]

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

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

R. Cressman, V. 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.  Google Scholar

[7]

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

[8]

J. Cui, Y. 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.  Google Scholar

[9]

J. Dockery, V. Hutson, K. 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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[14]

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

[15]

Y. Kang, S. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

M. Kummel, D. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[29]

J. Passarge, S. Hol, M. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[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.   Google Scholar

Figure 1.  Different regions of $ \mu $ to examine the criterion $ F(\alpha_1, \alpha_2)>0 $ (when $ \xi>1 $)
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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
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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
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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 $
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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 $
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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
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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
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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
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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
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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. $
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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. $
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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. $
[1]

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