Periodic solutions of third order predator-prey equations simulating an immune response

1974 ◽  
Vol 55 (2) ◽  
pp. 93-123 ◽  
Author(s):  
George H. Pimbley
Keyword(s):  
Immune Response ◽  
Periodic Solutions ◽  
Third Order ◽  
Predator Prey

Existence and stability of periodic solutions of a third-order non-linear autonomous system simulating immune response in animals

1977 ◽  
Vol 77 (1-2) ◽  
pp. 163-175 ◽  
Author(s):  
In-Ding Hsü ◽  
Nicholas D. Kazarinoff
Keyword(s):  
Immune Response ◽  
Steady State ◽  
Periodic Solutions ◽  
Stability Criterion ◽  
Autonomous System ◽  
Third Order ◽  
Non Linear ◽  
Existence And Stability ◽  
Linear Autonomous System ◽  
Non Linear System

SynopsisA 3 × 3 autonomous, non-linear system of ordinary differential equations modelling the immune response in animals to invasion by active self-replicating antigens has been introduced by G. I. Bell and studied by G. H. Pimbley Jr. Using Hopf's theorem on bifurcating periodic solutions and a stability criterion of Hsu and Kazarinoff, we obtain existence of a family of unstable periodic solutions bifurcating from one steady state of a reduced 2×2 form of the 3×3 system. We show that no periodic solutions bifurcate from the other steady state. We also prove existence and exhibit a stability criterion for families of periodic solutions of the full 3×3 system. We provide two numerical examples. The second shows existence of orbitally stable families of periodic solutions of the 3×3 system.

scholarly journals Positive Periodic Solutions of Third-Order Ordinary Differential Equations with Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Yongxiang Li ◽  
Qiang Li
Keyword(s):  
Periodic Solutions ◽  
Fixed Point Index ◽  
Index Theory ◽  
Existence Results ◽  
Positive Periodic Solutions ◽  
Point Index ◽  
Third Order ◽  
Order Ordinary Differential Equation ◽  
The Third ◽  
Fixed Point Index Theory

The existence results of positiveω-periodic solutions are obtained for the third-order ordinary differential equation with delaysu′′′(t)+a(t)u(t)=f(t,u(t-τ0),u′(t-τ1),u′′(t-τ2)),t∈ℝ,wherea∈C(ℝ,(0,∞))isω-periodic function andf:ℝ×[0,∞)×ℝ2→[0,∞)is a continuous function which isω-periodic int,and τ0,τ1,τ2are positive constants. The discussion is based on the fixed-point index theory in cones.

scholarly journals Existence and Lyapunov Stability of Positive Periodic Solutions for a Third-Order Neutral Differential Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-28 ◽  
Author(s):  
Jingli Ren ◽  
Zhibo Cheng ◽  
Yueli Chen
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Periodic Solutions ◽  
Lyapunov Stability ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Neutral Differential Equation ◽  
Positive Periodic Solutions ◽  
Third Order

By applying Green's function of third-order differential equation and a fixed point theorem in cones, we obtain some sufficient conditions for existence, nonexistence, multiplicity, and Lyapunov stability of positive periodic solutions for a third-order neutral differential equation.

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