## The Annals of Applied Probability

### From stochastic, individual-based models to the canonical equation of adaptive dynamics in one step

#### Abstract

We consider a model for Darwinian evolution in an asexual population with a large but nonconstant populations size characterized by a natural birth rate, a logistic death rate modeling competition and a probability of mutation at each birth event. In the present paper, we study the long-term behavior of the system in the limit of large population ($K\to\infty$) size, rare mutations ($u\to0$) and small mutational effects ($\sigma\to0$), proving convergence to the canonical equation of adaptive dynamics (CEAD). In contrast to earlier works, for example, by Champagnat and Méléard, we take the three limits simultaneously, that is, $u=u_{K}$ and $\sigma=\sigma_{K}$, tend to zero with $K$, subject to conditions that ensure that the time-scale of birth and death events remains separated from that of successful mutational events. This slows down the dynamics of the microscopic system and leads to serious technical difficulties that require the use of completely different methods. In particular, we cannot use the law of large numbers on the diverging time needed for fixation to approximate the stochastic system with the corresponding deterministic one. To solve this problem, we develop a “stochastic Euler scheme” based on coupling arguments that allows to control the time evolution of the stochastic system over time-scales that diverge with $K$.

#### Article information

Source
Ann. Appl. Probab. Volume 27, Number 2 (2017), 1093-1170.

Dates
Revised: February 2016
First available in Project Euclid: 26 May 2017

https://projecteuclid.org/euclid.aoap/1495764375

Digital Object Identifier
doi:10.1214/16-AAP1227

#### Citation

Baar, Martina; Bovier, Anton; Champagnat, Nicolas. From stochastic, individual-based models to the canonical equation of adaptive dynamics in one step. Ann. Appl. Probab. 27 (2017), no. 2, 1093--1170. doi:10.1214/16-AAP1227. https://projecteuclid.org/euclid.aoap/1495764375

#### References

• [1] Aldous, D. and Fill, J. Reversible Markov chains and random walks on graphs. In progress. Manuscript available at https://www.stat.berkeley.edu/~aldous/RWG/book.pdf.
• [2] Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Die Grundlehren der Mathematischen Wissenschaften 196. Springer, New York.
• [3] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
• [4] Bogachev, V. I. (2007). Measure Theory. Vol. I, II. Springer, Berlin.
• [5] Bovier, A. and den Hollander, F. (2015). Metastability: A Potential-Theoretic Approach. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 351. Springer, Cham.
• [6] Champagnat, N. (2006). A microscopic interpretation for adaptive dynamics trait substitution sequence models. Stochastic Process. Appl. 116 1127–1160.
• [7] Champagnat, N., Ferrière, R. and Ben Arous, G. (2001). The canonical equation of adaptive dynamics: A mathematical view. Selection 2 73–83.
• [8] Champagnat, N., Ferrière, R. and Méléard, S. (2008). From individual stochastic processes to macroscopic models in adaptive evolution. Stoch. Models 24 2–44.
• [9] Champagnat, N. and Méléard, S. (2011). Polymorphic evolution sequence and evolutionary branching. Probab. Theory Related Fields 151 45–94.
• [10] Dieckmann, U. and Law, R. (1996). The dynamical theory of coevolution: A derivation from stochastic ecological processes. J. Math. Biol. 34 579–612.
• [11] Durinx, M., Metz, J. A. J. and Meszéna, G. (2008). Adaptive dynamics for physiologically structured population models. J. Math. Biol. 56 673–742.
• [12] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
• [13] Fournier, N. and Méléard, S. (2004). A microscopic probabilistic description of a locally regulated population and macroscopic approximations. Ann. Appl. Probab. 14 1880–1919.
• [14] Freidlin, M. I. and Wentzell, A. D. (2012). Random Perturbations of Dynamical Systems, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 260. Springer, Heidelberg.
• [15] Geritz, S. A. H. (2005). Resident-invader dynamics and the coexistence of similar strategies. J. Math. Biol. 50 67–82.
• [16] Hofbauer, J. and Sigmund, K. (1990). Adaptive dynamics and evolutionary stability. Appl. Math. Lett. 3 75–79.
• [17] Istas, J. (2005). Mathematical Modeling for the Life Sciences. Springer, Berlin.
• [18] Jagers, P. and Lagerås, A. N. (2008). General branching processes conditioned on extinction are still branching processes. Electron. Commun. Probab. 13 540–547.
• [19] Metz, J. (2008). Fitness. Encyclopedia of Ecology 2 1599–1612.
• [20] Metz, J., Nisbet, R. and Geritz, S. (1992). How should we define “fitness” for general ecological scenarios? Trends Ecol. Evol. 7 198–202.
• [21] Metz, J. A. J., Geritz, S. A. H., Meszéna, G., Jacobs, F. J. A. and van Heerwaarden, J. S. (1996). Adaptive dynamics, a geometrical study of the consequences of nearly faithful reproduction. In Stochastic and Spatial Structures of Dynamical Systems (Amsterdam, 1995). Konink. Nederl. Akad. Wetensch. Verh. Afd. Natuurk. Eerste Reeks 45 183–231. North-Holland, Amsterdam.
• [22] Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York.