# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2022038
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## A spatially and size-structured population model with unbounded birth process

 Department of Mathematics, Faculty of Sciences, Ibn Zohr University, Hay Dakhla, BP8106, 80000–Agadir, Morocco

Received  June 2021 Revised  January 2022 Early access March 2022

In this paper, we consider a spatially and size structured population model with unbounded birth process. Firstly, the model is transformed into a closed-loop system, and hence the well-posedness is established by using the feedback theory of regular linear systems. Moreover, the solution to the resulting closed-loop system is given by a perturbed semigroup. Secondly, we give a condition on birth and death rates in such a way that the solution decays exponentially. To do this, we show that the semigroup solution is positive and hence we derive a characterization of exponential stability due to the technique tools of positive semigroups. We mention that our results extend a previous work in [D. Yan and X. Fu, Comm. Pure Appl. Anal. 15 (2016), 637–655] to the unbounded situation.

Citation: Abed Boulouz. A spatially and size-structured population model with unbounded birth process. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022038
##### References:
 [1] D. M. Auslander, G. F. Oster and C. B. Huffaker, Dynamics of interacting populations, J. Franklin Inst., 297 (1974), 345-376. [2] G. D. Blasio, Nonlinear age-dependent population growth with history-dependent birth rate, Math. Biosci., 46 (1979), 279-291.  doi: 10.1016/0025-5564(79)90073-7. [3] A. Boulouz, H. Bounit and S. Hadd, Feedback theory approach to positivity and stability of evolution equations, Syst & Control Lett., 161 (2022) 105167. [4] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Systems, New York, Springer-Verlag, 2000. [5] J. Z. Farkas, Stability conditions for a nonlinear size-structured model, Nonlinear Anal. Real World Appl., 6 (2005), 962-969.  doi: 10.1016/j.nonrwa.2004.06.002. [6] J. Z. Farkas and T. Hagen, Stability and regularity results for a size-structured population model, J. Math. Anal. Appl., 328 (2007), 119-136.  doi: 10.1016/j.jmaa.2006.05.032. [7] G. Greiner, A typical Perron–Frobenius theorem with applications to an age-dependent population equation, Infinite-dimensional systems, 1076 (1984), 86-100.  doi: 10.1007/BFb0072769. [8] G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213-229. [9] B. Guo and W. Chan, A semigroup approach to age dependent population dynamics with time delay, Comm. Partial Differential Equations, 14 (1989), 809-832.  doi: 10.1080/03605308908820630. [10] M. Gyllenberg and G. F. Webb, Asynchronous exponential growth of semigroups of nonlinear operators, J. Math. Anal. Appl., 167 (1992), 443-467.  doi: 10.1016/0022-247X(92)90218-3. [11] S. Hadd, R. Manzo and A. Rhandi, Unbounded perturbations of the generator domain, Discr. Cont. Dyn. Syst., 35 (2015), 703-723.  doi: 10.3934/dcds.2015.35.703. [12] M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Giardini Editori, Pisa, 1994. [13] Z. Liu, P. Magal and H. Tang, Hopf bifurcation for a spatially and age structured population dynamics model, Discr. Cont. Dyn. Syst. B, 20 (2015), 1735-1757.  doi: 10.3934/dcdsb.2015.20.1735. [14] Z. Mei and J. G. Peng, Dynamic boundary systems with boundary feedback and population system with unbounded birth process, Math. Meth. Appl. Sci., 38 (2015), 1642-1651.  doi: 10.1002/mma.3175. [15] S. Pizzera, An age dependent population equation with delayed birth process, Math. Meth. Appl. Sci., 27 (2004), 427-439.  doi: 10.1002/mma.462. [16] S. Pizzera and L. Tonetto, Asynchronous exponential growth for an age dependent population equation with delayed birth process, J. Evol. Equ., 5 (2005), 61-77.  doi: 10.1007/s00028-004-0159-6. [17] A. Rhandi and R. Schnaubelt, Asymptotic behaviour of a non-autonomous population equation with diffusion in L1, Discr. Cont. Dyn. Syst., 5 (1999), 663-683.  doi: 10.3934/dcds.1999.5.663. [18] W. E. Ricker, Computation and interpretation of biological studies of fish populations, Bull. Fish. Res. Board Can., 191 (1975). [19] D. Salamon, Infinite-dimensional linear system with unbounded control and observation: A functional analytic approach, Trans. Amer. Math. Soc., 300 (1987), 383-431.  doi: 10.2307/2000351. [20] J. O. Staffans, Well-posed Linear Systems, Encyclopedia of Mathematics and its Applications, 103, Cambridge: Cambridge University Press, 2005. doi: 10.1017/CBO9780511543197. [21] G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Monographs and Textbooks in Pure and Applied Mathematics, vol. 89, Marcel Dekker, New York, 1985. [22] L. Weis, The stability of positive semigroups on $L^p$-spaces, Proc. Amer. Math. Soc., 123 (1995), 3089-3094.  doi: 10.2307/2160665. [23] G. Weiss, Regular linear systems with feedback, Math. Control Signals Systems, 7 (1994), 23-57.  doi: 10.1007/BF01211484. [24] D. Yan and X. Fu, Asymptotic analysis a spatially and size-structured population dynamics model with delayed birth process, Comm. Pure Appl. Anal., 15 (2016), 637-655.  doi: 10.3934/cpaa.2016.15.637.

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##### References:
 [1] D. M. Auslander, G. F. Oster and C. B. Huffaker, Dynamics of interacting populations, J. Franklin Inst., 297 (1974), 345-376. [2] G. D. Blasio, Nonlinear age-dependent population growth with history-dependent birth rate, Math. Biosci., 46 (1979), 279-291.  doi: 10.1016/0025-5564(79)90073-7. [3] A. Boulouz, H. Bounit and S. Hadd, Feedback theory approach to positivity and stability of evolution equations, Syst & Control Lett., 161 (2022) 105167. [4] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Systems, New York, Springer-Verlag, 2000. [5] J. Z. Farkas, Stability conditions for a nonlinear size-structured model, Nonlinear Anal. Real World Appl., 6 (2005), 962-969.  doi: 10.1016/j.nonrwa.2004.06.002. [6] J. Z. Farkas and T. Hagen, Stability and regularity results for a size-structured population model, J. Math. Anal. Appl., 328 (2007), 119-136.  doi: 10.1016/j.jmaa.2006.05.032. [7] G. Greiner, A typical Perron–Frobenius theorem with applications to an age-dependent population equation, Infinite-dimensional systems, 1076 (1984), 86-100.  doi: 10.1007/BFb0072769. [8] G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213-229. [9] B. Guo and W. Chan, A semigroup approach to age dependent population dynamics with time delay, Comm. Partial Differential Equations, 14 (1989), 809-832.  doi: 10.1080/03605308908820630. [10] M. Gyllenberg and G. F. Webb, Asynchronous exponential growth of semigroups of nonlinear operators, J. Math. Anal. Appl., 167 (1992), 443-467.  doi: 10.1016/0022-247X(92)90218-3. [11] S. Hadd, R. Manzo and A. Rhandi, Unbounded perturbations of the generator domain, Discr. Cont. Dyn. Syst., 35 (2015), 703-723.  doi: 10.3934/dcds.2015.35.703. [12] M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Giardini Editori, Pisa, 1994. [13] Z. Liu, P. Magal and H. Tang, Hopf bifurcation for a spatially and age structured population dynamics model, Discr. Cont. Dyn. Syst. B, 20 (2015), 1735-1757.  doi: 10.3934/dcdsb.2015.20.1735. [14] Z. Mei and J. G. Peng, Dynamic boundary systems with boundary feedback and population system with unbounded birth process, Math. Meth. Appl. Sci., 38 (2015), 1642-1651.  doi: 10.1002/mma.3175. [15] S. Pizzera, An age dependent population equation with delayed birth process, Math. Meth. Appl. Sci., 27 (2004), 427-439.  doi: 10.1002/mma.462. [16] S. Pizzera and L. Tonetto, Asynchronous exponential growth for an age dependent population equation with delayed birth process, J. Evol. Equ., 5 (2005), 61-77.  doi: 10.1007/s00028-004-0159-6. [17] A. Rhandi and R. Schnaubelt, Asymptotic behaviour of a non-autonomous population equation with diffusion in L1, Discr. Cont. Dyn. Syst., 5 (1999), 663-683.  doi: 10.3934/dcds.1999.5.663. [18] W. E. Ricker, Computation and interpretation of biological studies of fish populations, Bull. Fish. Res. Board Can., 191 (1975). [19] D. Salamon, Infinite-dimensional linear system with unbounded control and observation: A functional analytic approach, Trans. Amer. Math. Soc., 300 (1987), 383-431.  doi: 10.2307/2000351. [20] J. O. Staffans, Well-posed Linear Systems, Encyclopedia of Mathematics and its Applications, 103, Cambridge: Cambridge University Press, 2005. doi: 10.1017/CBO9780511543197. [21] G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Monographs and Textbooks in Pure and Applied Mathematics, vol. 89, Marcel Dekker, New York, 1985. [22] L. Weis, The stability of positive semigroups on $L^p$-spaces, Proc. Amer. Math. Soc., 123 (1995), 3089-3094.  doi: 10.2307/2160665. [23] G. Weiss, Regular linear systems with feedback, Math. Control Signals Systems, 7 (1994), 23-57.  doi: 10.1007/BF01211484. [24] D. Yan and X. Fu, Asymptotic analysis a spatially and size-structured population dynamics model with delayed birth process, Comm. Pure Appl. Anal., 15 (2016), 637-655.  doi: 10.3934/cpaa.2016.15.637.
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