American Institute of Mathematical Sciences

• Previous Article
Modeling multiple taxis: Tumor invasion with phenotypic heterogeneity, haptotaxis, and unilateral interspecies repellence
• DCDS-B Home
• This Issue
• Next Article
On a singularly perturbed semi-linear problem with Robin boundary conditions
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  January 2021 Early access  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 and Continuous Dynamical Systems - B, 2021, 26 (1) : 415-441. doi: 10.3934/dcdsb.2020298
References:
 [1] E. Beretta, F. 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. [2] A. Campbell, Conditions for the existence of bacteriophage, Evolution, 15 (1961), 153-165.  doi: 10.1111/j.1558-5646.1961.tb03139.x. [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. [4] C. Chicone, Ordinary Differential Equations with Applications, Texts in Applied Mathematics, 34, Springer-Verlag, New York, 1999. [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. [6] M. H. Cortez and J. S. Weitz, Coevolution can reverse predator-prey cycles, PNAS, 111 (2014), 7486-7491.  doi: 10.1073/pnas.1317693111. [7] J. O. Haerter, N. 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. [8] D. H. Janzen, Herbivores and the number of tree species in tropical forests, American Naturalist, 104 (1970), 501-528.  doi: 10.1086/282687. [9] L. F. Jover, M. 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. [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. [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. [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. [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. [14] B. R. Levin, F. 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. [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. [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. [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. [18] A. R. Stenholm, I. 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. [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. [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. [21] J. S. Weitz, Quantitative Viral Ecology: Dynamics of Viruses and Their Microbial Hosts, Monographs in Population Biology, Princeton University Press, 2015. [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.

show all references

References:
 [1] E. Beretta, F. 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. [2] A. Campbell, Conditions for the existence of bacteriophage, Evolution, 15 (1961), 153-165.  doi: 10.1111/j.1558-5646.1961.tb03139.x. [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. [4] C. Chicone, Ordinary Differential Equations with Applications, Texts in Applied Mathematics, 34, Springer-Verlag, New York, 1999. [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. [6] M. H. Cortez and J. S. Weitz, Coevolution can reverse predator-prey cycles, PNAS, 111 (2014), 7486-7491.  doi: 10.1073/pnas.1317693111. [7] J. O. Haerter, N. 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. [8] D. H. Janzen, Herbivores and the number of tree species in tropical forests, American Naturalist, 104 (1970), 501-528.  doi: 10.1086/282687. [9] L. F. Jover, M. 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. [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. [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. [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. [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. [14] B. R. Levin, F. 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. [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. [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. [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. [18] A. R. Stenholm, I. 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. [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. [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. [21] J. S. Weitz, Quantitative Viral Ecology: Dynamics of Viruses and Their Microbial Hosts, Monographs in Population Biology, Princeton University Press, 2015. [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.
The projection of the phase diagram of model (1) onto the $N_1N_2$ plane. Left: $\beta = 11.5$; right: $\beta = 20$
The time series of model (1). Left: $\beta = 11.5$; right: $\beta = 20$
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 $E_0^{nnv}=(0,0,0)$ $r_1 <\omega$, $r_2 <\omega$ $E_1^{nnv}=(\tilde{N}_1,0,0)$ $r_1 > \omega$ $r_1 >r_2$, $N_1^\ast >\tilde{N}_1$ $E_2^{nnv}=(0,\tilde{N}_2,0)$ $r_2 > \omega$ $r_1 \tilde{N}_2$ $E_3^{nnv}=(N_1^\ast,0,\tilde{V}^\ast)$ $N_1^\ast <\tilde{N}_1$ ($r_1 >\omega$ 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 $E_0^{nnv}=(0,0,0)$ $r_1 <\omega$, $r_2 <\omega$ $E_1^{nnv}=(\tilde{N}_1,0,0)$ $r_1 > \omega$ $r_1 >r_2$, $N_1^\ast >\tilde{N}_1$ $E_2^{nnv}=(0,\tilde{N}_2,0)$ $r_2 > \omega$ $r_1 \tilde{N}_2$ $E_3^{nnv}=(N_1^\ast,0,\tilde{V}^\ast)$ $N_1^\ast <\tilde{N}_1$ ($r_1 >\omega$ 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) $\left(\frac{\phi_1}{\phi_2}-\frac{r_1}{r_2}\right)(N_2^\ast-\eta) <0$ $\left(\frac{\phi_1}{\phi_2}-\frac{r_1}{r_2}\right)(r_1-r_2) >0$ $E_5^{nnv}=(N_1^c,N_2^c,V^c)$ $(N_1^\ast-\eta)\left(\frac{\phi_1}{\phi_2}-\frac{\beta_2}{\beta_1}\right) <0$ $\left(\frac{\phi_1}{\phi_2}-\frac{r_1}{r_2}\right)\left(\frac{\phi_1}{\phi_2}-\frac{\beta_2}{\beta_1}\right) >0$ $(N_2^\ast-\eta)\left(\frac{\phi_1}{\phi_2}-\frac{\beta_2}{\beta_1}\right) >0$
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) $r_2 <\omega \tilde{N}_1$ U GAS - - - - (c) $r_2 <\omega \tilde{N}_2$ U - GAS - - - (e) $r_1 <\omega \omega, N_1^\ast <\tilde{N}_1, N_2^\ast <\tilde{N}_2 \\ (\phi_1r_2-\phi_2r_1)(N_1^\ast-\eta) <0 \\ (\phi_1r_2-\phi_2r_1)(N_2^\ast-\eta) <0 \end{array}$ 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) $\omega \tilde{N}_2$ 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) $\omega \tilde{N}_1$, $N_2^\ast <\tilde{N}_2$ 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) $\omega \tilde{N}_1$, $N_2^\ast >\tilde{N}_2$ U U GAS - - - (n) $\begin{array}{l} \mbox{(a) }r_1 >r_2 >\omega, \phi_1 r_2 >\phi_2r_1, \\ \tilde{N}_1 >N_1^\ast >\eta >N_2^\ast, \tilde{N}_2 >N_2^\ast; \\ \mbox{or (b) } \omega r_2 >\omega, \phi_1 r_2 >\phi_2r_1, \\ N_1^\ast <\eta \tilde{N}_1 >N_1^\ast >\eta >N_2^\ast \end{array}$ 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) $\begin{array}{l} \omega r_2 >\omega, \phi_1 r_2 >\phi_2r_1, \\ N_1^\ast <\eta <\tilde{N}_2 <\tilde{N}_1, \tilde{N}_2 \tilde{N}_1, \tilde{N}_2 >\tilde{N}_1 >\eta >N_2^\ast \end{array}$ 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) $r_2 <\omega \tilde{N}_1$ U GAS - - - - (c) $r_2 <\omega \tilde{N}_2$ U - GAS - - - (e) $r_1 <\omega \omega, N_1^\ast <\tilde{N}_1, N_2^\ast <\tilde{N}_2 \\ (\phi_1r_2-\phi_2r_1)(N_1^\ast-\eta) <0 \\ (\phi_1r_2-\phi_2r_1)(N_2^\ast-\eta) <0 \end{array}$ 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) $\omega \tilde{N}_2$ 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) $\omega \tilde{N}_1$, $N_2^\ast <\tilde{N}_2$ 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) $\omega \tilde{N}_1$, $N_2^\ast >\tilde{N}_2$ U U GAS - - - (n) $\begin{array}{l} \mbox{(a) }r_1 >r_2 >\omega, \phi_1 r_2 >\phi_2r_1, \\ \tilde{N}_1 >N_1^\ast >\eta >N_2^\ast, \tilde{N}_2 >N_2^\ast; \\ \mbox{or (b) } \omega r_2 >\omega, \phi_1 r_2 >\phi_2r_1, \\ N_1^\ast <\eta \tilde{N}_1 >N_1^\ast >\eta >N_2^\ast \end{array}$ 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) $\begin{array}{l} \omega r_2 >\omega, \phi_1 r_2 >\phi_2r_1, \\ N_1^\ast <\eta <\tilde{N}_2 <\tilde{N}_1, \tilde{N}_2 \tilde{N}_1, \tilde{N}_2 >\tilde{N}_1 >\eta >N_2^\ast \end{array}$ U U U - U S
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 $E_0=(0,0,0,0)$ $r_1 <\omega$, $r_2 <\omega$ $E_1=(\tilde{N}_1,0,0,0)$ $r_1 > \omega$ $r_1 >r_2$, $N_{1,1}^\ast >\tilde{N}_1$, $N_{1,2}^\ast >\tilde{N}_1$ $E_2=(0,\tilde{N}_2,0,0)$ $r_2 > \omega$ $r_1 \tilde{N}_2$, $N_{2,2}^\ast >\tilde{N}_2$ $E_3=(N_{1,1}^\ast,0,\frac{r_1(\tilde{N}_1-N_{1,1}^\ast)}{K\phi_{11}},0)$ $N_{1,1}^\ast <\tilde{N}_1$ $\begin{array}{l} B\Phi_3 >0 \\ \Phi R_1\cdot (N_{1,1}^\ast-\eta_1) >0 \end{array}$ $E_4=(N_{1,2}^\ast,0,0,\frac{r_1(\tilde{N}_1-N_{1,2}^\ast)}{K\phi_{12}})$ $N_{1,2}^\ast <\tilde{N}_1$ $\begin{array}{l} B\Phi_3 <0 \\ \Phi R_2\cdot (N_{1,2}^\ast-\eta_2) >0 \end{array}$ $E_5=(0,N_{2,1}^\ast,\frac{r_2(\tilde{N}_2-N_{2,1}^\ast)}{K\phi_{21}},0)$ $N_{2,1}^\ast <\tilde{N}_2$ $\begin{array}{l} B\Phi_4 >0 \\ \Phi R_1\cdot (N_{2,1}^\ast-\eta_1) <0 \end{array}$ $E_6=(0,N_{2,2}^\ast,0,\frac{r_2(\tilde{N}_2-N_{2,2}^\ast)}{K\phi_{22}})$ $N_{2,2}^\ast <\tilde{N}_2$ $\begin{array}{l} B\Phi_4 <0 \\ \Phi R_2\cdot(N_{2,2}^\ast-\eta_2) <0 \end{array}$ $E_7=(N_1^c,N_2^c,V_1^c,0)$ $\begin{array}{l} (N_{2,1}^\ast-\eta_1)\cdot B\Phi_1 >0 \\ (N_{1,1}^\ast-\eta_1)\cdot B\Phi_1 <0 \\ (r_1-r_2)\Phi R_1 >0 \end{array}$ $\begin{array}{l} NN <0 \\ \Phi R_1\cdot B\Phi_1 >0 \end{array}$ $E_8=(\hat{N}_1^c,\hat{N}_2^c,0,\hat{V}_2^c)$ $\begin{array}{l} (N_{2,2}^\ast-\eta_2)\cdot B\Phi_2 >0 \\ (N_{1,2}^\ast-\eta_2)\cdot B\Phi_2 <0 \\ (r_1-r_2)\Phi R_2 >0 \end{array}$ $\begin{array}{l} NN_h <0 \\ \Phi R_2\cdot B\Phi_2 >0 \end{array}$ $E_9=(N_1^p,N_2^p,V_1^p,V_2^p)$ $\begin{array}{l}B\Phi\cdot B\Phi_3 <0 \\ B\Phi\cdot B\Phi_4 >0 \\ \Phi R_1\cdot B\Phi_1\cdot NN\cdot B\Phi \cdot \Phi\Phi >0 \\ \Phi R_2\cdot B\Phi_2\cdot NN_h\cdot B\Phi \cdot \Phi\Phi >0 \end{array}$ $\begin{array}{l} B\Phi \cdot \Phi\Phi >0 \\ (14) \end{array}$
 Equilibrium Existence condition Stability condition $E_0=(0,0,0,0)$ $r_1 <\omega$, $r_2 <\omega$ $E_1=(\tilde{N}_1,0,0,0)$ $r_1 > \omega$ $r_1 >r_2$, $N_{1,1}^\ast >\tilde{N}_1$, $N_{1,2}^\ast >\tilde{N}_1$ $E_2=(0,\tilde{N}_2,0,0)$ $r_2 > \omega$ $r_1 \tilde{N}_2$, $N_{2,2}^\ast >\tilde{N}_2$ $E_3=(N_{1,1}^\ast,0,\frac{r_1(\tilde{N}_1-N_{1,1}^\ast)}{K\phi_{11}},0)$ $N_{1,1}^\ast <\tilde{N}_1$ $\begin{array}{l} B\Phi_3 >0 \\ \Phi R_1\cdot (N_{1,1}^\ast-\eta_1) >0 \end{array}$ $E_4=(N_{1,2}^\ast,0,0,\frac{r_1(\tilde{N}_1-N_{1,2}^\ast)}{K\phi_{12}})$ $N_{1,2}^\ast <\tilde{N}_1$ $\begin{array}{l} B\Phi_3 <0 \\ \Phi R_2\cdot (N_{1,2}^\ast-\eta_2) >0 \end{array}$ $E_5=(0,N_{2,1}^\ast,\frac{r_2(\tilde{N}_2-N_{2,1}^\ast)}{K\phi_{21}},0)$ $N_{2,1}^\ast <\tilde{N}_2$ $\begin{array}{l} B\Phi_4 >0 \\ \Phi R_1\cdot (N_{2,1}^\ast-\eta_1) <0 \end{array}$ $E_6=(0,N_{2,2}^\ast,0,\frac{r_2(\tilde{N}_2-N_{2,2}^\ast)}{K\phi_{22}})$ $N_{2,2}^\ast <\tilde{N}_2$ $\begin{array}{l} B\Phi_4 <0 \\ \Phi R_2\cdot(N_{2,2}^\ast-\eta_2) <0 \end{array}$ $E_7=(N_1^c,N_2^c,V_1^c,0)$ $\begin{array}{l} (N_{2,1}^\ast-\eta_1)\cdot B\Phi_1 >0 \\ (N_{1,1}^\ast-\eta_1)\cdot B\Phi_1 <0 \\ (r_1-r_2)\Phi R_1 >0 \end{array}$ $\begin{array}{l} NN <0 \\ \Phi R_1\cdot B\Phi_1 >0 \end{array}$ $E_8=(\hat{N}_1^c,\hat{N}_2^c,0,\hat{V}_2^c)$ $\begin{array}{l} (N_{2,2}^\ast-\eta_2)\cdot B\Phi_2 >0 \\ (N_{1,2}^\ast-\eta_2)\cdot B\Phi_2 <0 \\ (r_1-r_2)\Phi R_2 >0 \end{array}$ $\begin{array}{l} NN_h <0 \\ \Phi R_2\cdot B\Phi_2 >0 \end{array}$ $E_9=(N_1^p,N_2^p,V_1^p,V_2^p)$ $\begin{array}{l}B\Phi\cdot B\Phi_3 <0 \\ B\Phi\cdot B\Phi_4 >0 \\ \Phi R_1\cdot B\Phi_1\cdot NN\cdot B\Phi \cdot \Phi\Phi >0 \\ \Phi R_2\cdot B\Phi_2\cdot NN_h\cdot B\Phi \cdot \Phi\Phi >0 \end{array}$ $\begin{array}{l} B\Phi \cdot \Phi\Phi >0 \\ (14) \end{array}$
 [1] Tao Feng, Zhipeng Qiu, Xinzhu Meng. Dynamics of a stochastic hepatitis C virus system with host immunity. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6367-6385. doi: 10.3934/dcdsb.2019143 [2] Hossein Mohebbi, Azim Aminataei, Cameron J. Browne, Mohammad Reza Razvan. Hopf bifurcation of an age-structured virus infection model. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 861-885. doi: 10.3934/dcdsb.2018046 [3] Cameron J. Browne, Sergei S. Pilyugin. Global analysis of age-structured within-host virus model. Discrete and Continuous Dynamical Systems - B, 2013, 18 (8) : 1999-2017. doi: 10.3934/dcdsb.2013.18.1999 [4] Shikun Wang. Dynamics of a chemostat system with two patches. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6261-6278. doi: 10.3934/dcdsb.2019138 [5] Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344 [6] Hui li, Manjun Ma. Global dynamics of a virus infection model with repulsive effect. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4783-4797. doi: 10.3934/dcdsb.2019030 [7] Tin Phan, Bruce Pell, Amy E. Kendig, Elizabeth T. Borer, Yang Kuang. Rich dynamics of a simple delay host-pathogen model of cell-to-cell infection for plant virus. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 515-539. doi: 10.3934/dcdsb.2020261 [8] Zhikun She, Xin Jiang. Threshold dynamics of a general delayed within-host viral infection model with humoral immunity and two modes of virus transmission. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3835-3861. doi: 10.3934/dcdsb.2020259 [9] Hossein Pourbashash, Sergei S. Pilyugin, Patrick De Leenheer, Connell McCluskey. Global analysis of within host virus models with cell-to-cell viral transmission. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3341-3357. doi: 10.3934/dcdsb.2014.19.3341 [10] Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735 [11] Stephen Pankavich, Nathan Neri, Deborah Shutt. Bistable dynamics and Hopf bifurcation in a refined model of early stage HIV infection. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 2867-2893. doi: 10.3934/dcdsb.2020044 [12] Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325 [13] Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051 [14] Yan-Xia Dang, Zhi-Peng Qiu, Xue-Zhi Li, Maia Martcheva. Global dynamics of a vector-host epidemic model with age of infection. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1159-1186. doi: 10.3934/mbe.2017060 [15] Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507 [16] Tewfik Sari, Frederic Mazenc. Global dynamics of the chemostat with different removal rates and variable yields. Mathematical Biosciences & Engineering, 2011, 8 (3) : 827-840. doi: 10.3934/mbe.2011.8.827 [17] Cuicui Jiang, Kaifa Wang, Lijuan Song. Global dynamics of a delay virus model with recruitment and saturation effects of immune responses. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1233-1246. doi: 10.3934/mbe.2017063 [18] Cuicui Jiang, Wendi Wang. Complete classification of global dynamics of a virus model with immune responses. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 1087-1103. doi: 10.3934/dcdsb.2014.19.1087 [19] Feng-Bin Wang, Junping Shi, Xingfu Zou. Dynamics of a host-pathogen system on a bounded spatial domain. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2535-2560. doi: 10.3934/cpaa.2015.14.2535 [20] Xiulan Lai, Xingfu Zou. A reaction diffusion system modeling virus dynamics and CTLs response with chemotaxis. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2567-2585. doi: 10.3934/dcdsb.2016061

2020 Impact Factor: 1.327