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# Influence of mutations in phenotypically-structured populations in time periodic environment

This work was partially supported by grants from Région Ile-de-France and Inria team Mamba

• We study a parabolic Lotka-Volterra equation, with an integral term representing competition, and time periodic growth rate. This model represents a trait structured population in a time periodic environment. After showing the convergence of the solution to the unique positive and periodic solution of the problem, we study the influence of different factors on the mean limit population. As this quantity is the opposite of a certain eigenvalue, we are able to investigate the influence of the diffusion rate, the period length and the time variance of the environment fluctuations. We also give biological interpretation of the results in the framework of cancer, if the model represents a cancerous cells population under the influence of a periodic treatment. In this framework, we show that the population might benefit from a intermediate rate of mutation.

Mathematics Subject Classification: 35B10, 35K57, 35R09, 92D15.

 Citation: • • Figure 1.  Proliferation and sensitivity functions chosen for the simulations

Figure 2.  Final mean populations $\bar{\rho}_N$ represented as functions of the mutation rate $D$ for various treatment schedules. The treatments are of the form $C(t) = \mathbf{1}_{0\leq t \leq \tau}M/\tau$, with $M = 2$ and $\tau$ varying between $1/4$ and $T = 2$. Populations $A$, $B$ and $C$ are detailed in figure 3

Figure 3.  Populations $A$, $B$ and $C$ of figure 2 are detailed for $\tau = 1/4$. We represent the population repartition in phenotypes just before treatment, just after treatment and the mean population

Figure 4.  Illustration of the conjecture that $\bar{\rho}_N$ is a decreasing function of the time of drug administration $\tau$. These numerical simulations were performed for $T = 2$

Figure 5.  Illustration of Proposition 4 on the influence of $T$ on the population size. These numerical simulations were performed for $\tau = T/2$ and a fixed diffusion coefficient $D = 0.1$

Figure 6.  Mean limit population $\bar{\rho}_N$ as a function of both diffusion and drug dosage per period $M$

Figure 7.  Populations for random changing of environment with same mean value

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