# American Institute of Mathematical Sciences

2016, 13(1): 19-41. doi: 10.3934/mbe.2016.13.19

## Discrete or distributed delay? Effects on stability of population growth

 1 CIMAB, University of Milano, via C. Saldini 50, I20133 Milano 2 Department of Mathematics and Computer Science, University of Udine, via delle Scienze 206, I33100 Udine

Received  March 2015 Revised  August 2015 Published  October 2015

The growth of a population subject to maturation delay is modeled by using either a discrete delay or a delay continuously distributed over the population. The occurrence of stability switches (stable-unstable-stable) of the positive equilibrium as the delay increases is investigated in both cases. Necessary and sufficient conditions are provided by analyzing the relevant characteristic equations. It is shown that for any choice of parameter values for which the discrete delay model presents stability switches there exists a maximum delay variance beyond which no switch occurs for the continuous delay model: the delay variance has a stabilizing effect. Moreover, it is illustrated how, in the presence of switches, the unstable delay domain is as larger as lower is the ratio between the juveniles and the adults mortality rates.
Citation: Edoardo Beretta, Dimitri Breda. Discrete or distributed delay? Effects on stability of population growth. Mathematical Biosciences & Engineering, 2016, 13 (1) : 19-41. doi: 10.3934/mbe.2016.13.19
##### References:
 [1] E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165. doi: 10.1137/S0036141000376086. [2] D. Breda, S. Maset and R. Vermiglio, Pseudospectral differencing methods for characteristic roots of delay differential equations, SIAM J. Sci. Comput., 27 (2005), 482-495. doi: 10.1137/030601600. [3] D. Breda, S. Maset and R. Vermiglio, Stability of Linear Delay Differential Equations - A Numerical Approach with MATLAB, Springer Briefs in Control, Automation and Robotics, Springer, New York, 2015. doi: 10.1007/978-1-4939-2107-2. [4] K. L. Cooke, P. van der Driessche and X. Zou, Interaction of maturation delay and nonlinear birth in population and epidemic models, J. Math. Biol., 39 (1999), 332-352. doi: 10.1007/s002850050194. [5] M. Hazewinkel, Encyclopedia of Mathematics, Springer, 2001. [6] Z. Jiang and W. Zhang, Bifurcation analysis in single-species population model with delay, Sci. China Math., 53 (2010), 1475-1481. doi: 10.1007/s11425-010-4008-5. [7] H. L. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, no. 57 in Texts in Applied Mathematics, Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8. [8] J. Wei and X. Zou, Bifurcation analysis of a population model and the resulting {SIS} epidemic model with delay, J. Comput. Appl. Math., 197 (2006), 169-187. doi: 10.1016/j.cam.2005.10.037.

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##### References:
 [1] E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165. doi: 10.1137/S0036141000376086. [2] D. Breda, S. Maset and R. Vermiglio, Pseudospectral differencing methods for characteristic roots of delay differential equations, SIAM J. Sci. Comput., 27 (2005), 482-495. doi: 10.1137/030601600. [3] D. Breda, S. Maset and R. Vermiglio, Stability of Linear Delay Differential Equations - A Numerical Approach with MATLAB, Springer Briefs in Control, Automation and Robotics, Springer, New York, 2015. doi: 10.1007/978-1-4939-2107-2. [4] K. L. Cooke, P. van der Driessche and X. Zou, Interaction of maturation delay and nonlinear birth in population and epidemic models, J. Math. Biol., 39 (1999), 332-352. doi: 10.1007/s002850050194. [5] M. Hazewinkel, Encyclopedia of Mathematics, Springer, 2001. [6] Z. Jiang and W. Zhang, Bifurcation analysis in single-species population model with delay, Sci. China Math., 53 (2010), 1475-1481. doi: 10.1007/s11425-010-4008-5. [7] H. L. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, no. 57 in Texts in Applied Mathematics, Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8. [8] J. Wei and X. Zou, Bifurcation analysis of a population model and the resulting {SIS} epidemic model with delay, J. Comput. Appl. Math., 197 (2006), 169-187. doi: 10.1016/j.cam.2005.10.037.
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