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Stationary distribution, ergodicity and extinction of a stochastic generalized logistic system. (English) Zbl 1261.60057
This paper studies the long term dynamics of the generalized logistic stochastic differential equation \[ dx_t=x_t (r-a x^{\theta})dt + \sum_{i=1}^{n}a_i x dB_i(t)+ \sum_{i=1}^{n}\beta_{i} x^{1+\theta}dB_{i}(t), \] where \(B_i\) are independent Brownian motions and \(a>0\), \(\theta>0\). In the absence of noise, this system is known to have a positive equilibrium which is globally asymptotically stable for \(r>0\). The paper examines the validity of this result in the presence of noise, where now the statement has to be modified in terms of the stationary distribution and its ergodicity properties. It is proved that, for small enough values of the noise which depend also on the values of the parameters of the deterministic system, there exists a stationary invariant distribution which is ergodic. Furthermore, conditions under which extinction is possible are provided. The results are based essentially on appropriately constructed Lyapunov functions and are supported by simulation.

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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