This paper presents a prey–predator model with a Holling type II functional response that incorporates fear and refuge effects in the prey population. The model assumes that fear-induced birth rate reduction occurs only in the refuge-seeking prey. The positivity and boundedness of the system are established, followed by analyses of the existence and stability (both local and global) of all equilibria. A Hopf bifurcation is shown to occur, leading to the emergence of a limit cycle. The existence of a heteroclinic orbit connecting two saddle points, as well as a heteroclinic loop, is also demonstrated when a unique positive equilibrium exists. Analytical and numerical investigations explore the impacts of fear, refuge, and predator aggressiveness, supported by bifurcation diagrams. The results show that fear enhances system stability and reduces predator density without affecting prey density at the coexisting equilibrium.
We study almost periodic solutions of neutral-type neural dynamic equations with bounded delays on an almost periodic time scale. The main operator-theoretic issue is the inversion of a genuinely multi-channel neutral operator (A_i x_i)(t)=x_i(t)-∑ _k=1^m_ic_ik(t)x_i(δ _ik(t)) on AP(𝕋) . Under the explicit norm-smallness condition η _i:=sup _t∈𝕋∑ _k=1^m_i|c_ik(t)|<1 , the inverse A_i^-1 is represented by a Neumann series and satisfies ‖ A_i^-1‖≤ (1-η _i)^-1 . A key structural point is that powers of the neutral operator contain all finite compositions of the neutral delays. Hence the invariant-space hypothesis needed for the Neumann series must control every finite word generated by the neutral delay family, not only the delayed terms displayed in the original equation. We combine this inverse with the exponentially stable leakage resolvent to obtain coefficient-level small-gain conditions for the existence and uniqueness of an almost periodic solution. Global exponential stability is then proved in a weighted phase norm by estimating the neutral identity itself, without imposing an additional pointwise inverse estimate for the state error. Continuous-time and discrete-time criteria, a numerical stability illustration, and a continuous–discrete hybrid example with nonconstant graininess are included to demonstrate the applicability of the abstract conditions.
This paper construct a new 5D hyperchaotic system with three positive Lyapunov exponents through merge a state feedback controller and coupling strategies to the unusual 3D chaotic system. The proposed system is relatively simple, consisting of only nine terms, especially when compared to other available systems. Some relevant features have been analyzed both theoretically and numerically, including phase portraits, equilibrium, dissipative, Lyapunov exponents, and Kaplan-Yorke dimension. Moreover, the software-based electronic circuit is implemented using the Multisim program and verified with an oscilloscope. The results demonstrate that the attractors produced by the electronic circuit closely consistent with those generated in MATLAB, highlighting the system’s efficiency and effectiveness. Finally, a color image encryption scheme was implemented using the new 5D system. Statistical tests and simulation results demonstrated that the proposed encryption method offers excellent encryption performance and high efficiency.
By a direct method it is proved the existence of sign-changing and non-negative ground states for the p-Schrödinger-Poisson system {[ -Δ _p u + V(x)|u|^p-2u + K(x)ϕ (x)|u|^p-2u = a(x)|u|^q-2u, x ∈ℝ^N,; -Δ _p ϕ = K(x)|u|^p, x ∈ℝ^N, ]. where N∈ℕ , 2p= -1+√(1+8N)
In this paper, we investigate the existence of ground state solutions for a class of periodic nonlinear Dirac–Maxwell systems by means of variational methods. Compared with the existing literature, the assumptions on the periodic potentials are substantially weakened, and the structural conditions imposed on the nonlinear terms are relaxed by replacing modulus-type assumptions with conditions involving the real part of the complex inner product. Under these weaker hypotheses, we establish the existence of ground state solutions, thereby extending the applicability of the corresponding existence theory to a broader class of periodic potentials and nonlinearities. In addition, we present an explicit illustrative example together with graphical representations to demonstrate the effectiveness and applicability of the proposed assumptions. The obtained results generalize and complement the corresponding results of Peng et al.
