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January  2021, 26(1): 415-441. doi: 10.3934/dcdsb.2020298

The dynamics of a two host-two virus system in a chemostat environment

1. 

Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan

2. 

Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588, USA

* Corresponding author: Sze-Bi Hsu

Received  June 2020 Revised  August 2020 Published  October 2020

Fund Project: The first author is supported by the grant MOST 108-2115-M-007-007. The second author is supported by NSF Grant DMS 1411703

The coevolution or coexistence of multiple viruses with multiple hosts has been an important issue in viral ecology. This paper is to study the mathematical properties of the solutions of a chemostat model for two host species and two virus species. By virtue of the global dynamics of its submodels and the theories of uniform persistence and Hopf bifurcation, we derive sufficient conditions for the coexistence of two hosts with two viruses and coexistence of two hosts with one virus, as well as occurrence of Hopf bifurcation.

Citation: Sze-Bi Hsu, Yu Jin. The dynamics of a two host-two virus system in a chemostat environment. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 415-441. doi: 10.3934/dcdsb.2020298
References:
[1]

E. BerettaF. Solimano and Y. Tang, Analysis of a chemostat model for bacteria and virulent bacteriophage, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 495-520.  doi: 10.3934/dcdsb.2002.2.495.  Google Scholar

[2]

A. Campbell, Conditions for the existence of bacteriophage, Evolution, 15 (1961), 153-165.  doi: 10.1111/j.1558-5646.1961.tb03139.x.  Google Scholar

[3]

L. Chao, B. R. Levin, and F. M. Stewart, A complex community in a simple habitat: An experimental study with bacteria and phage, Ecology, 58 (1977), 369-378. doi: 10.2307/1935611.  Google Scholar

[4]

C. Chicone, Ordinary Differential Equations with Applications, Texts in Applied Mathematics, 34, Springer-Verlag, New York, 1999.  Google Scholar

[5]

J. H. Connell, On the role of natural enemies in preventing competitive exclusion in some marine animals and in rain forest trees, in Dynamics of Populations: Proceedings of the Advanced Study Institute on Dynamics of Numbers in Populations, Oosterbeek, Netherlands, 1971. Google Scholar

[6]

M. H. Cortez and J. S. Weitz, Coevolution can reverse predator-prey cycles, PNAS, 111 (2014), 7486-7491.  doi: 10.1073/pnas.1317693111.  Google Scholar

[7]

J. O. HaerterN. Mitarai and K. Sneppen, Phage and bacteria support mutual diversity in a narrowing staircase of coexistence, ISME Journal, 8 (2014), 2317-2326.  doi: 10.1038/ismej.2014.80.  Google Scholar

[8]

D. H. Janzen, Herbivores and the number of tree species in tropical forests, American Naturalist, 104 (1970), 501-528.  doi: 10.1086/282687.  Google Scholar

[9]

L. F. JoverM. H. Cortez and J. S. Weitz, Mechanisms of multi-strain coexistence in host-phage systems with nested infection networks, J. Theoret. Biol., 332 (2013), 65-77.  doi: 10.1016/j.jtbi.2013.04.011.  Google Scholar

[10]

D. A. Korytowski and H. L. Smith, A special class of Lotka-Volterra models of bacteria-virus infection networks, in Applied Analysis in Biological and Physical Sciences, Springer Proc. Math. Stat., 186, Springer, New Delhi, 2016,113–119. doi: 10.1007/978-81-322-3640-5_7.  Google Scholar

[11]

D. A. Korytowski and H. L. Smith, How nested and monogamous infection networks in host-phage communities come to be, Theoretical Ecology, 8 (2015), 111-120.  doi: 10.1007/s12080-014-0236-6.  Google Scholar

[12]

D. A. Korytowski and H. Smith, Permanence and stability of a kill the winner model in marine ecology, Bull. Math. Biol., 79 (2017), 995-1004.  doi: 10.1007/s11538-017-0265-6.  Google Scholar

[13]

R. E. Lenski and B. R. Levin, Constraints on the coevolution of bacteria and virulent phage: A model, some experiments, and predictions for natural communities, American Naturalist, 125 (1985), 585-602.  doi: 10.1086/284364.  Google Scholar

[14]

B. R. LevinF. M. Stewart and L. Chao, Resource-limited growth, competition, and predation: A model and experimental studies with bacteria and bacteriophage, American Naturalist, 111 (1977), 3-24.  doi: 10.1086/283134.  Google Scholar

[15]

J. Mallet, The struggle for existence: How the notion of carrying capacity, K, obscures the links between demography, Darwinian evolution, and speciation, Evol. Ecology Res., 14 (2012), 627-665.   Google Scholar

[16]

H. L. Smith and H. R. Thieme, Persistence of bacteria and phages in a chemostat, J. Math. Biol., 64 (2012), 951-979.  doi: 10.1007/s00285-011-0434-4.  Google Scholar

[17]

H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2.  Google Scholar

[18]

A. R. StenholmI. Dalsgaard and M. Middelboe, Isolation and characterization of bacteriophages infecting the fish pathogen Flavobacterium psychrophilum, Appl. Environ. Microbiol., 74 (2008), 4070-4078.  doi: 10.1128/AEM.00428-08.  Google Scholar

[19]

T. F. Thingstad, Elements of a theory for the mechanisms controlling abundance, diversity, and biogeochemical role of lytic bacterial viruses in aquatic systems, Limnology and Oceanography, 45 (2000), 1320-1328.  doi: 10.4319/lo.2000.45.6.1320.  Google Scholar

[20]

T. Thingstad and R. Lignell, Theoretical models for the control of bacterial growth rate, abundance, diversity and carbon demand, Aquatic Microbial Ecology, 13 (1997), 19-27.  doi: 10.3354/ame013019.  Google Scholar

[21] J. S. Weitz, Quantitative Viral Ecology: Dynamics of Viruses and Their Microbial Hosts, Monographs in Population Biology, Princeton University Press, 2015.   Google Scholar
[22]

X.-Q. Zhao, Dynamical Systems in Population Biology, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 16, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

show all references

References:
[1]

E. BerettaF. Solimano and Y. Tang, Analysis of a chemostat model for bacteria and virulent bacteriophage, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 495-520.  doi: 10.3934/dcdsb.2002.2.495.  Google Scholar

[2]

A. Campbell, Conditions for the existence of bacteriophage, Evolution, 15 (1961), 153-165.  doi: 10.1111/j.1558-5646.1961.tb03139.x.  Google Scholar

[3]

L. Chao, B. R. Levin, and F. M. Stewart, A complex community in a simple habitat: An experimental study with bacteria and phage, Ecology, 58 (1977), 369-378. doi: 10.2307/1935611.  Google Scholar

[4]

C. Chicone, Ordinary Differential Equations with Applications, Texts in Applied Mathematics, 34, Springer-Verlag, New York, 1999.  Google Scholar

[5]

J. H. Connell, On the role of natural enemies in preventing competitive exclusion in some marine animals and in rain forest trees, in Dynamics of Populations: Proceedings of the Advanced Study Institute on Dynamics of Numbers in Populations, Oosterbeek, Netherlands, 1971. Google Scholar

[6]

M. H. Cortez and J. S. Weitz, Coevolution can reverse predator-prey cycles, PNAS, 111 (2014), 7486-7491.  doi: 10.1073/pnas.1317693111.  Google Scholar

[7]

J. O. HaerterN. Mitarai and K. Sneppen, Phage and bacteria support mutual diversity in a narrowing staircase of coexistence, ISME Journal, 8 (2014), 2317-2326.  doi: 10.1038/ismej.2014.80.  Google Scholar

[8]

D. H. Janzen, Herbivores and the number of tree species in tropical forests, American Naturalist, 104 (1970), 501-528.  doi: 10.1086/282687.  Google Scholar

[9]

L. F. JoverM. H. Cortez and J. S. Weitz, Mechanisms of multi-strain coexistence in host-phage systems with nested infection networks, J. Theoret. Biol., 332 (2013), 65-77.  doi: 10.1016/j.jtbi.2013.04.011.  Google Scholar

[10]

D. A. Korytowski and H. L. Smith, A special class of Lotka-Volterra models of bacteria-virus infection networks, in Applied Analysis in Biological and Physical Sciences, Springer Proc. Math. Stat., 186, Springer, New Delhi, 2016,113–119. doi: 10.1007/978-81-322-3640-5_7.  Google Scholar

[11]

D. A. Korytowski and H. L. Smith, How nested and monogamous infection networks in host-phage communities come to be, Theoretical Ecology, 8 (2015), 111-120.  doi: 10.1007/s12080-014-0236-6.  Google Scholar

[12]

D. A. Korytowski and H. Smith, Permanence and stability of a kill the winner model in marine ecology, Bull. Math. Biol., 79 (2017), 995-1004.  doi: 10.1007/s11538-017-0265-6.  Google Scholar

[13]

R. E. Lenski and B. R. Levin, Constraints on the coevolution of bacteria and virulent phage: A model, some experiments, and predictions for natural communities, American Naturalist, 125 (1985), 585-602.  doi: 10.1086/284364.  Google Scholar

[14]

B. R. LevinF. M. Stewart and L. Chao, Resource-limited growth, competition, and predation: A model and experimental studies with bacteria and bacteriophage, American Naturalist, 111 (1977), 3-24.  doi: 10.1086/283134.  Google Scholar

[15]

J. Mallet, The struggle for existence: How the notion of carrying capacity, K, obscures the links between demography, Darwinian evolution, and speciation, Evol. Ecology Res., 14 (2012), 627-665.   Google Scholar

[16]

H. L. Smith and H. R. Thieme, Persistence of bacteria and phages in a chemostat, J. Math. Biol., 64 (2012), 951-979.  doi: 10.1007/s00285-011-0434-4.  Google Scholar

[17]

H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2.  Google Scholar

[18]

A. R. StenholmI. Dalsgaard and M. Middelboe, Isolation and characterization of bacteriophages infecting the fish pathogen Flavobacterium psychrophilum, Appl. Environ. Microbiol., 74 (2008), 4070-4078.  doi: 10.1128/AEM.00428-08.  Google Scholar

[19]

T. F. Thingstad, Elements of a theory for the mechanisms controlling abundance, diversity, and biogeochemical role of lytic bacterial viruses in aquatic systems, Limnology and Oceanography, 45 (2000), 1320-1328.  doi: 10.4319/lo.2000.45.6.1320.  Google Scholar

[20]

T. Thingstad and R. Lignell, Theoretical models for the control of bacterial growth rate, abundance, diversity and carbon demand, Aquatic Microbial Ecology, 13 (1997), 19-27.  doi: 10.3354/ame013019.  Google Scholar

[21] J. S. Weitz, Quantitative Viral Ecology: Dynamics of Viruses and Their Microbial Hosts, Monographs in Population Biology, Princeton University Press, 2015.   Google Scholar
[22]

X.-Q. Zhao, Dynamical Systems in Population Biology, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 16, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

Figure 1.  The projection of the phase diagram of model (1) onto the $ N_1N_2 $ plane. Left: $ \beta = 11.5 $; right: $ \beta = 20 $
Figure 2.  The time series of model (1). Left: $ \beta = 11.5 $; right: $ \beta = 20 $
Table 1.  The conditions for existence and local stability of equilibria of (4). Here, an equilibrium exists means it is nonnegative for $ E_1^{nnv} $-$ E_4^{nnv} $ and positive for $ E_5^{nnv} $
Equilibrium Existence condition Stability condition
$ E_0^{nnv}=(0,0,0) $ $ r_1 <\omega $, $ r_2 <\omega $
$ E_1^{nnv}=(\tilde{N}_1,0,0) $ $ r_1 > \omega $ $ r_1 >r_2 $, $ N_1^\ast >\tilde{N}_1 $
$ E_2^{nnv}=(0,\tilde{N}_2,0) $ $ r_2 > \omega $ $ r_1 <r_2 $, $ N_2^\ast >\tilde{N}_2 $
$ E_3^{nnv}=(N_1^\ast,0,\tilde{V}^\ast) $ $ N_1^\ast <\tilde{N}_1 $ ($ r_1 >\omega $ required) $ \left(\frac{\phi_1}{\phi_2}-\frac{r_1}{r_2}\right)(N_1^\ast-\eta) >0 $
$ E_4^{nnv}=(0,N_2^\ast,V^\ast) $ $ N_2^\ast <\tilde{N}_2 $ ($ r_2 >\omega $ required) $ \left(\frac{\phi_1}{\phi_2}-\frac{r_1}{r_2}\right)(N_2^\ast-\eta) <0 $
$ \left(\frac{\phi_1}{\phi_2}-\frac{r_1}{r_2}\right)(r_1-r_2) >0 $
$ E_5^{nnv}=(N_1^c,N_2^c,V^c) $ $ (N_1^\ast-\eta)\left(\frac{\phi_1}{\phi_2}-\frac{\beta_2}{\beta_1}\right) <0 $ $ \left(\frac{\phi_1}{\phi_2}-\frac{r_1}{r_2}\right)\left(\frac{\phi_1}{\phi_2}-\frac{\beta_2}{\beta_1}\right) >0 $
$ (N_2^\ast-\eta)\left(\frac{\phi_1}{\phi_2}-\frac{\beta_2}{\beta_1}\right) >0 $
Equilibrium Existence condition Stability condition
$ E_0^{nnv}=(0,0,0) $ $ r_1 <\omega $, $ r_2 <\omega $
$ E_1^{nnv}=(\tilde{N}_1,0,0) $ $ r_1 > \omega $ $ r_1 >r_2 $, $ N_1^\ast >\tilde{N}_1 $
$ E_2^{nnv}=(0,\tilde{N}_2,0) $ $ r_2 > \omega $ $ r_1 <r_2 $, $ N_2^\ast >\tilde{N}_2 $
$ E_3^{nnv}=(N_1^\ast,0,\tilde{V}^\ast) $ $ N_1^\ast <\tilde{N}_1 $ ($ r_1 >\omega $ required) $ \left(\frac{\phi_1}{\phi_2}-\frac{r_1}{r_2}\right)(N_1^\ast-\eta) >0 $
$ E_4^{nnv}=(0,N_2^\ast,V^\ast) $ $ N_2^\ast <\tilde{N}_2 $ ($ r_2 >\omega $ required) $ \left(\frac{\phi_1}{\phi_2}-\frac{r_1}{r_2}\right)(N_2^\ast-\eta) <0 $
$ \left(\frac{\phi_1}{\phi_2}-\frac{r_1}{r_2}\right)(r_1-r_2) >0 $
$ E_5^{nnv}=(N_1^c,N_2^c,V^c) $ $ (N_1^\ast-\eta)\left(\frac{\phi_1}{\phi_2}-\frac{\beta_2}{\beta_1}\right) <0 $ $ \left(\frac{\phi_1}{\phi_2}-\frac{r_1}{r_2}\right)\left(\frac{\phi_1}{\phi_2}-\frac{\beta_2}{\beta_1}\right) >0 $
$ (N_2^\ast-\eta)\left(\frac{\phi_1}{\phi_2}-\frac{\beta_2}{\beta_1}\right) >0 $
Table 2.  Global or local dynamics of (4). $ E_0^{nnv} $-$ E_5^{nnv} $ are defined in (5). Conditions for $ E_5^{nnv} $ to be positive or not may not be all listed. "-" represents that some compartments of the equilibrium are negative. "U" represents "unstable"; "GAS" represents "globally asymptotically stable", "S" represents "locally asymptotically stable"
Condition $ E_0^{nnv} $ $ E_1^{nnv} $ $ E_2^{nnv} $ $ E_3^{nnv} $ $ E_4^{nnv} $ $ E_5^{nnv} $
(a) $ r_1 <\omega $, $ r_2 <\omega $ GAS - - - - -
(b) $ r_2 <\omega <r_1 $, $ N_1^\ast >\tilde{N}_1 $ U GAS - - - -
(c) $ r_2 <\omega <r_1 $, $ N_1^\ast <\tilde{N}_1 $ U U - GAS - -
(d) $ r_1 <\omega <r_2 $, $ N_2^\ast >\tilde{N}_2 $ U - GAS - - -
(e) $ r_1 <\omega <r_2 $, $ N_2^\ast <\tilde{N}_2 $ U - U - GAS -
(f) $ \begin{array}{l} r_1,r_2 >\omega, N_1^\ast <\tilde{N}_1, N_2^\ast <\tilde{N}_2 \\ (\phi_1r_2-\phi_2r_1)(N_1^\ast-\eta) <0 \\ (\phi_1r_2-\phi_2r_1)(N_2^\ast-\eta) <0 \end{array} $ U U U U GAS -
(g) $ \begin{array}{l} r_1,r_2 >\omega, , N_1^\ast <\tilde{N}_1, N_2^\ast <\tilde{N}_2 \\ (\phi_1r_2-\phi_2r_1)(N_1^\ast-\eta) >0 \\ (\phi_1r_2-\phi_2r_1)(N_2^\ast-\eta) >0 \end{array} $ U U U GAS U -
(h) $ r_1 >r_2 >\omega $, $ N_1^\ast >\tilde{N}_1 $, $ N_2^\ast <\tilde{N}_2 $ U GAS U - U -
(i) $ \omega <r_1 <r_2 $, $ N_1^\ast <\tilde{N}_1 $, $ N_2^\ast >\tilde{N}_2 $ U U GAS U - -
(j) $ r_1 >r_2 >\omega $, $ N_1^\ast <\tilde{N}_1 $, $ N_2^\ast >\tilde{N}_2 $ U U U GAS - -
(k) $ \omega <r_1 <r_2 $, $ N_1^\ast >\tilde{N}_1 $, $ N_2^\ast <\tilde{N}_2 $ U U U - GAS -
(l) $ r_1 >r_2 >\omega $, $ N_1^\ast >\tilde{N}_1 $, $ N_2^\ast >\tilde{N}_2 $ U GAS U - - -
(m) $ \omega <r_1 <r_2 $, $ N_1^\ast >\tilde{N}_1 $, $ N_2^\ast >\tilde{N}_2 $ U U GAS - - -
(n) $ \begin{array}{l} \mbox{(a) }r_1 >r_2 >\omega, \phi_1 r_2 >\phi_2r_1, \\ \tilde{N}_1 >N_1^\ast >\eta >N_2^\ast, \tilde{N}_2 >N_2^\ast; \\ \mbox{or (b) } \omega <r_1 <r_2, \phi_1 r_2 <\phi_2r_1, \\ N_1^\ast <\eta <N_2^\ast <\tilde{N}_2, N_1^\ast <\tilde{N}_1 \end{array} $ U U U S S U
(o) $ \begin{array}{l} \mbox{(a) } r_1 >r_2 >\omega, \phi_1 r_2 >\phi_2r_1, \\ N_1^\ast <\eta <N_2^\ast <\tilde{N}_2 <\tilde{N}_1; \\ \mbox{or (b) } \omega <r_1 <r_2, \phi_1 r_2 <\phi_2r_1, \\ \tilde{N}_2 >\tilde{N}_1 >N_1^\ast >\eta >N_2^\ast \end{array} $ U U U U U S
(p) $ r_1 >r_2 >\omega $, $ \phi_1 r_2 >\phi_2r_1 $, U S U - S U
$ N_1^\ast >\tilde{N}_1 >\tilde{N}_2 >\eta >N_2^\ast $
(q) $ \begin{array}{l} \omega <r_1 <r_2, \phi_1 r_2 <\phi_2r_1, \\ N_1^\ast <\eta <\tilde{N}_1 <\tilde{N}_2 <N_2^\ast\end{array} $ U U S S - U
(r) $ \begin{array}{l} r_1 >r_2 >\omega, \phi_1 r_2 >\phi_2r_1, \\ N_1^\ast <\eta <\tilde{N}_2 <\tilde{N}_1, \tilde{N}_2 <N_2^\ast \end{array} $ U U U U - S
(s) $ \begin{array}{l} \omega <r_1 <r_2, \phi_1 r_2 <\phi_2r_1, \\ N_1^\ast >\tilde{N}_1, \tilde{N}_2 >\tilde{N}_1 >\eta >N_2^\ast \end{array} $ U U U - U S
Condition $ E_0^{nnv} $ $ E_1^{nnv} $ $ E_2^{nnv} $ $ E_3^{nnv} $ $ E_4^{nnv} $ $ E_5^{nnv} $
(a) $ r_1 <\omega $, $ r_2 <\omega $ GAS - - - - -
(b) $ r_2 <\omega <r_1 $, $ N_1^\ast >\tilde{N}_1 $ U GAS - - - -
(c) $ r_2 <\omega <r_1 $, $ N_1^\ast <\tilde{N}_1 $ U U - GAS - -
(d) $ r_1 <\omega <r_2 $, $ N_2^\ast >\tilde{N}_2 $ U - GAS - - -
(e) $ r_1 <\omega <r_2 $, $ N_2^\ast <\tilde{N}_2 $ U - U - GAS -
(f) $ \begin{array}{l} r_1,r_2 >\omega, N_1^\ast <\tilde{N}_1, N_2^\ast <\tilde{N}_2 \\ (\phi_1r_2-\phi_2r_1)(N_1^\ast-\eta) <0 \\ (\phi_1r_2-\phi_2r_1)(N_2^\ast-\eta) <0 \end{array} $ U U U U GAS -
(g) $ \begin{array}{l} r_1,r_2 >\omega, , N_1^\ast <\tilde{N}_1, N_2^\ast <\tilde{N}_2 \\ (\phi_1r_2-\phi_2r_1)(N_1^\ast-\eta) >0 \\ (\phi_1r_2-\phi_2r_1)(N_2^\ast-\eta) >0 \end{array} $ U U U GAS U -
(h) $ r_1 >r_2 >\omega $, $ N_1^\ast >\tilde{N}_1 $, $ N_2^\ast <\tilde{N}_2 $ U GAS U - U -
(i) $ \omega <r_1 <r_2 $, $ N_1^\ast <\tilde{N}_1 $, $ N_2^\ast >\tilde{N}_2 $ U U GAS U - -
(j) $ r_1 >r_2 >\omega $, $ N_1^\ast <\tilde{N}_1 $, $ N_2^\ast >\tilde{N}_2 $ U U U GAS - -
(k) $ \omega <r_1 <r_2 $, $ N_1^\ast >\tilde{N}_1 $, $ N_2^\ast <\tilde{N}_2 $ U U U - GAS -
(l) $ r_1 >r_2 >\omega $, $ N_1^\ast >\tilde{N}_1 $, $ N_2^\ast >\tilde{N}_2 $ U GAS U - - -
(m) $ \omega <r_1 <r_2 $, $ N_1^\ast >\tilde{N}_1 $, $ N_2^\ast >\tilde{N}_2 $ U U GAS - - -
(n) $ \begin{array}{l} \mbox{(a) }r_1 >r_2 >\omega, \phi_1 r_2 >\phi_2r_1, \\ \tilde{N}_1 >N_1^\ast >\eta >N_2^\ast, \tilde{N}_2 >N_2^\ast; \\ \mbox{or (b) } \omega <r_1 <r_2, \phi_1 r_2 <\phi_2r_1, \\ N_1^\ast <\eta <N_2^\ast <\tilde{N}_2, N_1^\ast <\tilde{N}_1 \end{array} $ U U U S S U
(o) $ \begin{array}{l} \mbox{(a) } r_1 >r_2 >\omega, \phi_1 r_2 >\phi_2r_1, \\ N_1^\ast <\eta <N_2^\ast <\tilde{N}_2 <\tilde{N}_1; \\ \mbox{or (b) } \omega <r_1 <r_2, \phi_1 r_2 <\phi_2r_1, \\ \tilde{N}_2 >\tilde{N}_1 >N_1^\ast >\eta >N_2^\ast \end{array} $ U U U U U S
(p) $ r_1 >r_2 >\omega $, $ \phi_1 r_2 >\phi_2r_1 $, U S U - S U
$ N_1^\ast >\tilde{N}_1 >\tilde{N}_2 >\eta >N_2^\ast $
(q) $ \begin{array}{l} \omega <r_1 <r_2, \phi_1 r_2 <\phi_2r_1, \\ N_1^\ast <\eta <\tilde{N}_1 <\tilde{N}_2 <N_2^\ast\end{array} $ U U S S - U
(r) $ \begin{array}{l} r_1 >r_2 >\omega, \phi_1 r_2 >\phi_2r_1, \\ N_1^\ast <\eta <\tilde{N}_2 <\tilde{N}_1, \tilde{N}_2 <N_2^\ast \end{array} $ U U U U - S
(s) $ \begin{array}{l} \omega <r_1 <r_2, \phi_1 r_2 <\phi_2r_1, \\ N_1^\ast >\tilde{N}_1, \tilde{N}_2 >\tilde{N}_1 >\eta >N_2^\ast \end{array} $ U U U - U S
Table 3.  The conditions for existence and stability of equilibria of model (1). Here, an equilibrium exists means it is nonnegative for $ E_1 $-$ E_8 $ and positive for $ E_9 $. The notations are defined in (9) and (10)
Equilibrium Existence condition Stability condition
$ E_0=(0,0,0,0) $ $ r_1 <\omega $, $ r_2 <\omega $
$ E_1=(\tilde{N}_1,0,0,0) $ $ r_1 > \omega $ $ r_1 >r_2 $, $ N_{1,1}^\ast >\tilde{N}_1 $, $ N_{1,2}^\ast >\tilde{N}_1 $
$ E_2=(0,\tilde{N}_2,0,0) $ $ r_2 > \omega $ $ r_1 <r_2 $, $ N_{2,1}^\ast >\tilde{N}_2 $, $ N_{2,2}^\ast >\tilde{N}_2 $
$ E_3=(N_{1,1}^\ast,0,\frac{r_1(\tilde{N}_1-N_{1,1}^\ast)}{K\phi_{11}},0) $ $ N_{1,1}^\ast <\tilde{N}_1 $ $ \begin{array}{l} B\Phi_3 >0 \\ \Phi R_1\cdot (N_{1,1}^\ast-\eta_1) >0 \end{array} $
$ E_4=(N_{1,2}^\ast,0,0,\frac{r_1(\tilde{N}_1-N_{1,2}^\ast)}{K\phi_{12}}) $ $ N_{1,2}^\ast <\tilde{N}_1 $ $ \begin{array}{l} B\Phi_3 <0 \\ \Phi R_2\cdot (N_{1,2}^\ast-\eta_2) >0 \end{array} $
$ E_5=(0,N_{2,1}^\ast,\frac{r_2(\tilde{N}_2-N_{2,1}^\ast)}{K\phi_{21}},0) $ $ N_{2,1}^\ast <\tilde{N}_2 $ $ \begin{array}{l} B\Phi_4 >0 \\ \Phi R_1\cdot (N_{2,1}^\ast-\eta_1) <0 \end{array} $
$ E_6=(0,N_{2,2}^\ast,0,\frac{r_2(\tilde{N}_2-N_{2,2}^\ast)}{K\phi_{22}}) $ $ N_{2,2}^\ast <\tilde{N}_2 $ $ \begin{array}{l} B\Phi_4 <0 \\ \Phi R_2\cdot(N_{2,2}^\ast-\eta_2) <0 \end{array} $
$ E_7=(N_1^c,N_2^c,V_1^c,0) $ $ \begin{array}{l} (N_{2,1}^\ast-\eta_1)\cdot B\Phi_1 >0 \\ (N_{1,1}^\ast-\eta_1)\cdot B\Phi_1 <0 \\ (r_1-r_2)\Phi R_1 >0 \end{array} $ $ \begin{array}{l} NN <0 \\ \Phi R_1\cdot B\Phi_1 >0 \end{array} $
$ E_8=(\hat{N}_1^c,\hat{N}_2^c,0,\hat{V}_2^c) $ $ \begin{array}{l} (N_{2,2}^\ast-\eta_2)\cdot B\Phi_2 >0 \\ (N_{1,2}^\ast-\eta_2)\cdot B\Phi_2 <0 \\ (r_1-r_2)\Phi R_2 >0 \end{array} $ $ \begin{array}{l} NN_h <0 \\ \Phi R_2\cdot B\Phi_2 >0 \end{array} $
$ E_9=(N_1^p,N_2^p,V_1^p,V_2^p) $ $ \begin{array}{l}B\Phi\cdot B\Phi_3 <0 \\ B\Phi\cdot B\Phi_4 >0 \\ \Phi R_1\cdot B\Phi_1\cdot NN\cdot B\Phi \cdot \Phi\Phi >0 \\ \Phi R_2\cdot B\Phi_2\cdot NN_h\cdot B\Phi \cdot \Phi\Phi >0 \end{array} $ $ \begin{array}{l} B\Phi \cdot \Phi\Phi >0 \\ (14) \end{array} $
Equilibrium Existence condition Stability condition
$ E_0=(0,0,0,0) $ $ r_1 <\omega $, $ r_2 <\omega $
$ E_1=(\tilde{N}_1,0,0,0) $ $ r_1 > \omega $ $ r_1 >r_2 $, $ N_{1,1}^\ast >\tilde{N}_1 $, $ N_{1,2}^\ast >\tilde{N}_1 $
$ E_2=(0,\tilde{N}_2,0,0) $ $ r_2 > \omega $ $ r_1 <r_2 $, $ N_{2,1}^\ast >\tilde{N}_2 $, $ N_{2,2}^\ast >\tilde{N}_2 $
$ E_3=(N_{1,1}^\ast,0,\frac{r_1(\tilde{N}_1-N_{1,1}^\ast)}{K\phi_{11}},0) $ $ N_{1,1}^\ast <\tilde{N}_1 $ $ \begin{array}{l} B\Phi_3 >0 \\ \Phi R_1\cdot (N_{1,1}^\ast-\eta_1) >0 \end{array} $
$ E_4=(N_{1,2}^\ast,0,0,\frac{r_1(\tilde{N}_1-N_{1,2}^\ast)}{K\phi_{12}}) $ $ N_{1,2}^\ast <\tilde{N}_1 $ $ \begin{array}{l} B\Phi_3 <0 \\ \Phi R_2\cdot (N_{1,2}^\ast-\eta_2) >0 \end{array} $
$ E_5=(0,N_{2,1}^\ast,\frac{r_2(\tilde{N}_2-N_{2,1}^\ast)}{K\phi_{21}},0) $ $ N_{2,1}^\ast <\tilde{N}_2 $ $ \begin{array}{l} B\Phi_4 >0 \\ \Phi R_1\cdot (N_{2,1}^\ast-\eta_1) <0 \end{array} $
$ E_6=(0,N_{2,2}^\ast,0,\frac{r_2(\tilde{N}_2-N_{2,2}^\ast)}{K\phi_{22}}) $ $ N_{2,2}^\ast <\tilde{N}_2 $ $ \begin{array}{l} B\Phi_4 <0 \\ \Phi R_2\cdot(N_{2,2}^\ast-\eta_2) <0 \end{array} $
$ E_7=(N_1^c,N_2^c,V_1^c,0) $ $ \begin{array}{l} (N_{2,1}^\ast-\eta_1)\cdot B\Phi_1 >0 \\ (N_{1,1}^\ast-\eta_1)\cdot B\Phi_1 <0 \\ (r_1-r_2)\Phi R_1 >0 \end{array} $ $ \begin{array}{l} NN <0 \\ \Phi R_1\cdot B\Phi_1 >0 \end{array} $
$ E_8=(\hat{N}_1^c,\hat{N}_2^c,0,\hat{V}_2^c) $ $ \begin{array}{l} (N_{2,2}^\ast-\eta_2)\cdot B\Phi_2 >0 \\ (N_{1,2}^\ast-\eta_2)\cdot B\Phi_2 <0 \\ (r_1-r_2)\Phi R_2 >0 \end{array} $ $ \begin{array}{l} NN_h <0 \\ \Phi R_2\cdot B\Phi_2 >0 \end{array} $
$ E_9=(N_1^p,N_2^p,V_1^p,V_2^p) $ $ \begin{array}{l}B\Phi\cdot B\Phi_3 <0 \\ B\Phi\cdot B\Phi_4 >0 \\ \Phi R_1\cdot B\Phi_1\cdot NN\cdot B\Phi \cdot \Phi\Phi >0 \\ \Phi R_2\cdot B\Phi_2\cdot NN_h\cdot B\Phi \cdot \Phi\Phi >0 \end{array} $ $ \begin{array}{l} B\Phi \cdot \Phi\Phi >0 \\ (14) \end{array} $
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