Distributed time-delay in Non-linear Population Models Manuel O. Cáceres Instituto Balseiro, Universidad Nacional de Cuyo, Centro Atómico Bariloche, CNEA, CONICET San Carlos de Bariloche, Argentina

Models of population dynamics with distributed time-delay. Which are the characteristics of these non-linear models, stabilities, relaxations, scaling times, etc.

Linear Stability Analysis (perturbations: n(t)~exp[(-c+iω)t]) around the fixed point: K*

Particular case: stability when the distributed time-delay G(s) is Exponential:

When G(s) is exponential the dynamics can be re-written as an ODE in 1+1 dimension Then it is possible to introduce an adiabatic elimination when epsilon is small ! (mean delay-time: <s>= ∫ s G(s) ds =1/λ)

Cases when G(s) are non-exponential can also be worked out Cases when G(s) are non-exponential can also be worked out. If there is a Green function associated to the pdf. For example the Laplace pdf :

1) When G(s) is the Laplace pdf the full dynamics can also be re-written as an ODE. 2) In this case the dimension of the dynamical system is increased in 1+2 dim. 3) As usual, the i.c. of the system can be extracted form its integral equation.

Example: A coupled model for the Logistic equation Consider the Logistic equation with a currying capacity “time-dependent” function K(t). Consider the delay Laplace pdf G(s) , then we can introduce a non-trivial distributed time-delay coupled carrying capacity model. In particular:

Stochastic models (remarks) Distributed-delay plus stochastic terms: an approach to the study of fluctuations. Form now on we will focus only in the exponential distributed time-delay model. Here for stochastic models we only consider external noise.

Stochastic approach in the exponential Distributed time-delay model (external noise) In the following (of the talk) we will adopt additive noise, with suitable B.C.

The Stochastic Adiabatic (Markov) Approach

Then to study the distributed delay system we only need to solve a one dim problem. The MFPT can be calculated in the Markov approximation. This is the characteristic time-scale in the pattern formation (θ —> K*) When the instability is linear and if the noise is small (θ~0) the MFPT is independent of the attractor (in its dominant θ-contribution) This is not case in a critical-point. A non-linear instability depends on the saturation!

The MFPT (0→Nf) in the adiabatic approximation (Markov approach) and for small noise is:

Solution of the problem in the Non-adiabatic approximation (the strong Non-Markov case)

Example of application: The distributed time-delay Logistic model with small additive stochastic perturbations. (mean delay time: ε= ∫t G(t) dt =1/λ)

Using Wiener integrals (SPPA), the MFPT (for small noise θ, and for any mean delay-time 1/λ) can be obtain analytically

Asymptotic results: In the regime λ>>1 we re-obtain the adiabatic result:

Monte Carlo Simulations with exponential time-delay. Theoretical calculation of the MFPT for any value of λ in the Logistic model with small stochastic perturbations (θ).

Summary and future extensions General theory Distributed time-delay in non-linear population models. Stochastic asymptotic dynamics (relaxation scaling-time). Generalization to extended systems (Fisher-like models, non-local stochastic PDE, etc.). Present applications Exponential distributed time-delay in Logistic models. Coupled Logistic-Capacity (complex time-delay models). The MFPT is the characteristic time-scale for the Pattern Formation. Initial State (unstable) Final State (attractor)

Thanks, for your attention… References: Theory 0-dim systems: MOC, JSP 156, 94 (2014) Monte Carlo: MOC & C.D. Rojas, Physica A 409, 61, (2014) Non-exponential delay distribution: MOC, PRE 90, 22137 (2014) SPPA with arbitrary noise: MOC, JSP 132, 487 (2008) Non-Local Fisher equation: M.A. Fuentes & MOC, Cent.Eur. J Phys 11,1623 (2013) Multiscale perturbation approach for extended systems: Math. Mod. Nat. Phen. (2015) Critical slowing-down in the MFPT for the non-local Fisher Equation: PRE, submitted (2015)

Stochastic Path Perturbation Approach (SPPA) for strong Non-Markov cases with small noise! The MFPT and its Distribution can analytically be calculated by perturbations on θ

There are two Green functions associated to the Laplace pdf :

On the ODE’s for the delay-distributed Logistic model. Using exponential G(s) we can go from 2 to 3 dim. From Laplace pdf distribution G(s) and taking tD →0