Abstract
Since population models are often subject to environmental noise, in this paper we stochastically perturb the Lotka-Volterra model with variable delay \(\dot{x}(t)=\text{diag}(x(t))\big{[}b+Ax(t)+Bx(t-\delta(t))\big{]}\) into the It\(\hat{\text{o}}\) form \(\text{d}x(t)=\text{diag}(x(t))\big{[}\big{(}b+Ax(t)+Bx(t-\delta(t))\big{)}\text{d}t+\sigma\text{d}w(t)\big{]}\). We show that under certain conditions, the deterministic delay equation and the perturbed delay equation have similar behaviour in the sense that both have positive solutions which will not explode to infinity in a finite time and, in fact, will be ultimately bounded.
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Xu, Y., Zhu, S., Hu, S. (2009). A Stochastic Lotka-Volterra Model with Variable Delay. In: Wang, H., Shen, Y., Huang, T., Zeng, Z. (eds) The Sixth International Symposium on Neural Networks (ISNN 2009). Advances in Intelligent and Soft Computing, vol 56. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01216-7_10
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DOI: https://doi.org/10.1007/978-3-642-01216-7_10
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