We study a discrete time model of the growth of two species of plankton with compet- itive and allelopathic effects on each other N1(k+1)= N1(k)exp{r1(k) − a11(k)N1(k) − a12(k)N2(k) − b1(k)N1(k)N2(k)}, N2(k +1 )= N2(k)exp{r2(k) − a21(k)N1(k) − a22(k) N2(k) − b2(k)N1(k)N2(k)} .As et of sufficient conditions is obtained for the existence of multiple positive periodic solutions for this model. The approach is based on Mawhin's continuation theorem of coincidence degree theory as well as some a priori estimates. Some new results are obtained.

This paper deals with the problem of cluster synchronization in networks with asymmetric negative couplings. By decomposing the coupling matrix into three matrices, and employing Lyapunov function method, sufficient conditions are derived for cluster synchronization. The conditions show that the couplings of multi-node clusters from one-node clusters have beneficial effects on cluster synchronization. Based on the effects of the one-node clusters, an effective and universal control scheme is put forward for the first time. The obtained results may help us better understand the relation between cluster synchronization and cluster structures of the networks. The validity of the control scheme is confirmed through two numerical simulations, in a network with no cluster structure and in a scale-free network.

In this paper, we investigate adaptive outer synchronization for identifying unknown network topology and parameters between two coupled complex networks with time-varying delays existing in the node dynamics and coupling forms. By designing adaptive controllers and updating laws, we obtain two theorems on the appearance of outer synchronization using Lyapunov stability theory. In the numerical examples, we provide the detailed identification criteria of network topology and system parameter according to the obtained theorems and give some numerical examples to show the efficiency of the proposed controllers.

All edges in the classical Watts and Strogatz's small-world network model are unweighted and cooperative (positive). By introducing competitive (negative) inter-cluster edges and assigning edge weights to mimic more realistic networks, this paper develops a modified model which possesses co-competitive weighted couplings and cluster structures while maintaining the common small-world network properties of small average shortest path lengths and large clustering coefficients. Based on theoretical analysis, it is proved that the new model with inter-cluster co-competition balance has an important dynamical property of robust cluster synchronous pattern formation. More precisely, clusters will neither merge nor split regardless of adding or deleting nodes and edges, under the condition of inter-cluster co-competition balance. Numerical simulations demonstrate the robustness of the model against the increase of the coupling strength and several topological variations.