# American Institute of Mathematical Sciences

• Previous Article
Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals
• DCDS-B Home
• This Issue
• Next Article
Gap solitons for the repulsive Gross-Pitaevskii equation with periodic potential: Coding and method for computation
June  2017, 22(4): 1231-1252. doi: 10.3934/dcdsb.2017060

## Stability analysis of an enteropathogen population growing within a heterogeneous group of animals

 Faculty of Sciences, Universitat Autónoma de Barcelona, 08193 Bellaterra, 08193 Bellaterra, Barcelona, Spain

* Corresponding authorr: carlesbarril@mat.uab.cat

Received  October 2015 Revised  November 2016 Published  February 2017

Fund Project: The first author is supported by Spanish Ministry of Education grant FPU13/04333.

An autonomous semi-linear hyperbolic pde system for the proliferation of bacteria within a heterogeneous population of animals is presented and analysed. It is assumed that bacteria grow inside the intestines and that they can be either attached to the epithelial wall or as free particles in the lumen. A condition involving ecological parameters is given, which can be used to decide the existence of endemic equilibria as well as local stability properties of the non-endemic one. Some implications on phage therapy are addressed.

Citation: Carles Barril, Àngel Calsina. Stability analysis of an enteropathogen population growing within a heterogeneous group of animals. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1231-1252. doi: 10.3934/dcdsb.2017060
##### References:

show all references

##### References:
Bifurcation diagram showing epidemic progression (dark regions) or eradication (white regions) in a system with two hosts. The changing parameters are the fraction of bacteriophage given to the first host ($q_0^1/(q_0^1+q_0^2)$ ranging from 0 to 1) and its detachment rate ($\delta_1$ ranging from 0 to 1.5). The bacterial and bacteriophage distributions along the intestine once the system has converged to the equilibria are shown for two different set of parameters A and B. The dashed line refers to attached bacteria while gray color is used for host one and black for host two. The other fix parameters used to do the numeric simulations are the total bacteriophage dose per time unit $q_0^1+q_0^2=11$ and: $c_h=l_h=1$, $\gamma_1^h(u)=1-u$, $\gamma_2^h(v)=1-v$, $\alpha_h =4$, $b=4$, $\kappa_1^h=0.06$, $\kappa_2^h=0.1$, $\lambda_1^h=\lambda_2^h=0.1$, $\mu_1=0.4$ and $\mu_2=0.1$ for all $h\in{1,2}$ and $\delta_2=0.5$. Notice that the two hosts only differ in their detachment rate ($\delta_1$ and $\delta_2$) and the treatment received ($q_0^1$ and $q_0^2$).
 [1] Janet Dyson, Rosanna Villella-Bressan, G.F. Webb. The steady state of a maturity structured tumor cord cell population. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 115-134. doi: 10.3934/dcdsb.2004.4.115 [2] Inom Mirzaev, David M. Bortz. A numerical framework for computing steady states of structured population models and their stability. Mathematical Biosciences & Engineering, 2017, 14 (4) : 933-952. doi: 10.3934/mbe.2017049 [3] Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735 [4] Leonid Shaikhet. Stability of a positive equilibrium state for a stochastically perturbed mathematical model of glassy-winged sharpshooter population. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1167-1174. doi: 10.3934/mbe.2014.11.1167 [5] Rinaldo M. Colombo, Mauro Garavello. Stability and optimization in structured population models on graphs. Mathematical Biosciences & Engineering, 2015, 12 (2) : 311-335. doi: 10.3934/mbe.2015.12.311 [6] Soohyun Bae. Weighted $L^\infty$ stability of positive steady states of a semilinear heat equation in $\R^n$. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 823-837. doi: 10.3934/dcds.2010.26.823 [7] Stéphane Mischler, Clément Mouhot. Stability, convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited granular media. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 159-185. doi: 10.3934/dcds.2009.24.159 [8] Luca Gerardo-Giorda, Pierre Magal, Shigui Ruan, Ousmane Seydi, Glenn Webb. Preface: Population dynamics in epidemiology and ecology. Discrete & Continuous Dynamical Systems - B, 2020, 25 (6) : ⅰ-ⅱ. doi: 10.3934/dcdsb.2020125 [9] Shangzhi Li, Shangjiang Guo. Dynamics of a stage-structured population model with a state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3523-3551. doi: 10.3934/dcdsb.2020071 [10] Federica Di Michele, Bruno Rubino, Rosella Sampalmieri. A steady-state mathematical model for an EOS capacitor: The effect of the size exclusion. Networks & Heterogeneous Media, 2016, 11 (4) : 603-625. doi: 10.3934/nhm.2016011 [11] Jitendra Kumar, Gurmeet Kaur, Evangelos Tsotsas. An accurate and efficient discrete formulation of aggregation population balance equation. Kinetic & Related Models, 2016, 9 (2) : 373-391. doi: 10.3934/krm.2016.9.373 [12] Dongxue Yan, Xianlong Fu. Asymptotic analysis of a spatially and size-structured population model with delayed birth process. Communications on Pure & Applied Analysis, 2016, 15 (2) : 637-655. doi: 10.3934/cpaa.2016.15.637 [13] Xianlong Fu, Dongmei Zhu. Stability analysis for a size-structured juvenile-adult population model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 391-417. doi: 10.3934/dcdsb.2014.19.391 [14] Xianlong Fu, Dongmei Zhu. Stability results for a size-structured population model with delayed birth process. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 109-131. doi: 10.3934/dcdsb.2013.18.109 [15] Hal L. Smith, Horst R. Thieme. Persistence and global stability for a class of discrete time structured population models. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4627-4646. doi: 10.3934/dcds.2013.33.4627 [16] La-Su Mai, Kaijun Zhang. Asymptotic stability of steady state solutions for the relativistic Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 981-1004. doi: 10.3934/dcds.2016.36.981 [17] Mei-hua Wei, Jianhua Wu, Yinnian He. Steady-state solutions and stability for a cubic autocatalysis model. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1147-1167. doi: 10.3934/cpaa.2015.14.1147 [18] Carmen Cortázar, Marta García-Huidobro, Pilar Herreros. On the uniqueness of bound state solutions of a semilinear equation with weights. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6761-6784. doi: 10.3934/dcds.2019294 [19] Kousuke Kuto. Stability and Hopf bifurcation of coexistence steady-states to an SKT model in spatially heterogeneous environment. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 489-509. doi: 10.3934/dcds.2009.24.489 [20] Yuxiang Li. Stabilization towards the steady state for a viscous Hamilton-Jacobi equation. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1917-1924. doi: 10.3934/cpaa.2009.8.1917

2019 Impact Factor: 1.27