Bifurcation Analysis and H ∞ Control of a Stochastic Competition Model with Time Delay

In this paper, we study a stochastic competition model with time delay and harvesting. We ﬁrst simplify it through the stochastic center manifold reduction principle and stochastic averaging method as a one-dimensional Markov diffusion process. Singular boundary theory and an invariant measure are applied to analyze stochastic stability and bifurcation. The T-S fuzzy model of the system is constructed, and the H ∞ fuzzy controller is designed to eliminate the bifurcation phenomenon through a linear matrix inequality approach. Numerical simulation is used to demonstrate our results.

The substitution of the elements of (Ψ , Φ) in the bilinear relation (8) yields the nonsingular matrix Then the basis Ψ ∈Ĉ is normalized tõ {N 4 cos ωs − N 2 sin ωs}, Substituting (Ψ (s), ϕ k (θ )), generated by the new inner product matrix (Ψ (s), Φ(θ )) in the bilinear pairing expression (8), the available identity matrix is Next, by defining z = (z 1 , z 2 ) = (Ψ , u t ), which actually represents the local coordinate system on the two-dimensional center manifold, induced by the basis Ψ , with the aid of equation (10) and (11), one can decompose u t into two parts to obtain which implies that the projection of u t on the center manifold is Φz. Then, applying (13) and (6) results in Thus, Finally, we obtain the equation of the stochastic center manifold: where N (z, ξ 1 , ξ 2 ) represents the nonlinear terms contributed from the original system to the stochastic center manifold.
Remark 1 If c r < −1, then the boundary r = +∞ is attractively natural, i.e., the system is unstable at the equilibrium point, and Hop f bifurcation is possible.

Hop f bifurcation
Based on (17), we can obtain the FPK equation as with the initial value condition where p(r,t|r 0 ,t 0 ) is the transition probability density of diffusion process r(t). The invariant measure of r(t) is the steady-state probability density p st (r), which is the solution of the degenerate system, as follows: Solving the above equation, one can obtain where h = 8µ 10 +µ 1 . We now calculate the most possible amplitude r * of system (17), i.e., p st (r) has a maximum value at r * . So, we have and the solution is r = 0 or r =r = √ −8µ 2 8µ 10 +µ 1 −µ 3 ( −8µ 2 8µ 10 +µ 1 −µ 3 < 1 2 ). The relations show that r =r. p st (r) is minimum at r = 0. Stochastic system (17) is almost unstable at r = 0. Hence, system (2) may show Hop f bifurcation at r =r. Then , (r =r).

H ∞ control
Hopf bifurcation indicates the instability of the system, which is not what we want to see. T-S fuzzy control [12] approximates or represents a global nonlinear system model using several local linear system models, and solves the control problem of the whole local nonlinear system by means of analysis and control of the linear system. H ∞ control based on T-S fuzzy control [13][14][15][16] is widely used in various fields. To eliminate this bifurcation phenomenon, the application of T-S fuzzy H ∞ control to solve the problem of nonlinear biological systems will be a general trend in the field of biological control [17,18].
Next, introducing the state feedback control for system (3), one can obtain the controlled system (19): where u(t) is the control variable.
We suppose that y i (t) ∈ [−k i , k i ], i = 1, 2, 3. A system of fuzzy equations is given as follows, which can describe system (19) when y i (t) ∈ [−k i , k i ], i = 1, 2, 3: . where ] T , h i (y(t)) = 1. The global model can be described as:ẏ (t + α)dα from the Newton-Leibniz formula, system (20) is equivalent tȯ Lemma 1 [19] For any u, v ∈ R n , and any matrix Z > 0 with appropriate dimensions, the following inequality holds: where * indicates a symmetric term, then u(t) = 2 ∑ j=1 h j (y(t))K j y(t) makes the H ∞ norm of system (23) less than γ. The closed-loop system is quadratically stable, , Proof Construct the singular Lyapunov function V (t) = y T (t)Py(t) +V 1 (t) +V 2 (t), After calculation, ] T . Then the derivative of the Lyapunov function is given from the above equations bẏ where Therefore, when Q < 0, the system (23)

Numerical analysis
We select some parameters to fully reflect the relationship among the steady-state probability density and position of stochastic Hop f bifurcation with the value of µ 2 in Section (3), and further demonstrate the effect of the controller in Section (4), with the following parameters: We know by calculation that when µ 10 = −0.6720, µ 1 = 2.2717, µ 3 = 0.7710, the four curves in Fig. 1 correspond to cond 1, 2, 3, 4, respectively, in Table 1. It can be seen from Table 1 The probabilities and the positions of the Hopf bifurcation occurrence.  The H ∞ controller is designed for system (19), so that the H ∞ norm of the closed-loop system (19) is less than γ = 0.01. Using LMI toolbox in MATLAB, the parameters satisfying Theorem 1 can be obtained as follows: ] . Fig. 2 shows that the output and state variables of the closed-loop system (19) gradually tend to zero over time. This fully demonstrates the feasibility of the controller.

Conclusion
This paper studied the Hop f bifurcation phenomenon of a stochastic competition population with stage structure under the influence of white noise. By using the stochastic center manifold and stochastic average method to reduce dimension, and through stability analysis and bifurcation position research, we can take µ 2 as a bifurcation parameter, and the position of stochastic Hop f bifurcation will increase with the increase of µ 2 , while the steady-state probability density of stochastic Hop f bifurcation will decrease. To eliminate the bifurcation phenomenon, a fuzzy state feedback controller was designed using the T-S fuzzy system control method to ensure the stability of the closed-loop system.