Lotka Volterra. Stability Analysis of Delayed Periodic Lotka Volterra Systems

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1 Lotka Volterra Stability Analysis of Delayed Periodic Lotka Volterra Systems 25 4

2 ii iii

3 Lotka Volterra 2 2 Lotka Volterra Lotka Volterra Lotka Volterra Lotka Volterra Lotka Volterra Schauder Lotka Volterra Lyapunov Lyapunov Lotka Volterra M O175.1, Q141.

4 Abstract The Lotka Volterra system is one of the most important models in mathematical ecology. Since it was proposed in the 192 s, successful applications of this model and its variants have been widely found in diverse areas of natural and social sciences, which have provided powerful tools for understanding the mechanisms of competition and dynamical evolution in a complex system. In recent years, the delayed periodic Lotka Volterra systems, which introduce the time delays and periodic oscillations into the original model, have attracted increasing interest. This thesis focus on the delayed periodic Lotka Volterra systems of a very general form. Using fundamental and straightforward analytic techniques, we discuss in detail the issues on positive and nonnegative periodic solutions that have significant ecological meanings. Several new results of existence and stability of periodic solutions are proved. This thesis is organized as follows. Chapter 1: Introduction. We briefly introduce the applications of Lotka Volterra systems in ecology, neural networks and other fields, and the necessity of research on delayed periodic Lotka Volterra systems. Model descriptions are then given, and challenging problems are proposed based on analyzing the ideas and methods currently used to investigate the model. Chapter 2: Existence and stability of positive periodic solutions. Positive periodic solutions of Lotka Volterra systems imply that all the species in an ecosystem coexist permanently with periodic oscillations. Taking a different way from those in the literature and applying Schauder s fixed point theorem, we prove some existence conditions for positive periodic solutions. And the global asymptotic stability and global exponential stability are also addressed. Chapter 3: Stability of nonnegative periodic solutions. The vanishing components of a stable periodic solution imply that the corresponding species are driven to extinction eventually. To study the stability of nonnegative periodic solutions, we first discuss the boundedness of solutions, and then prove the stability theorems by simple mathematical analysis. By using this approach, we avoid the possible difficulty in applying the Lyapunov s method.

5 iv Chapter 4: A numerical example and discussions. The comprehensive example is carefully designed and intended both to illustrate how to verify the stability criteria obtained in previous chapters, and to explain the distinction between the conditions for positive and nonnegative periodic solutions. All these results provide insights into the understanding of dynamical behavior of complex ecosystems. Keywords: Lotka Volterra system, neural network, delay, periodic solution, global asymptotic stability, global exponential stability, nonsingular M-matrix.

6 1.1 (trophic web) (1) (2) ( ) (3) ( ) (predator), (prey), Volterra ]. dn = N(a bp ), dt (1.1) dp dt = P (cn d), N(t) P (t) t a, b, c, d 192 Lotka 12] Lotka Volterra (). (1.1) n ( ) dx i dt = x i b i a ij x j, i = 1, 2,..., n, x i (t) i t b i i a ii > a ij (i j) Lotka Volterra ( ) 5, 19].

7 2 14, 17, 18, 22] Fukai Tanaka Lotka Volterra 7]. Asai MOS 2]. (recurrent) Lotka Volterra 17]. Lotka Volterra Lotka Volterra 8, 9, 21, 23] 2 9 Lotka Volterra Hopfield 1, 13, 32]. Lotka Volterra 1.2 n Lotka Volterra

8 3 dx i (t) = x i (t) b i (t) dt a ij (t)x j (t) x j (t s)d s µ ij (t, s) ] i = 1, 2,..., n,, (1.2) b i (t) a ij (t) ω > a ii (t) >, t, d s µ ij (t, s) d s µ ij (t + ω, s) = d s µ ij (t, s), t K ij (s) d s µ ij (t, s) dk ij (s) dk ij (s) < +, s dk ij (s) < + (1.3) i, j = 1, 2,..., n x i (s) = φ i (s), s (, ], (1.4) φ i (s) s φ i (s) >. d s µ ij (t, s) = b ij (t)δ(s + τ ij )ds + c ij (t, s)ds, δ( ) Dirac δ i, j = 1, 2,..., n, b ij (t) τ ij (t) ω c ij (t + ω, s) = c ij (t, s), τ ij (t), (1.2) dx i (t) = x i (t) b i (t) a ij (t)x j (t) b ij (t)x j (t τ ij (t)) dt ] c ij (t, s)x j (t s)ds, i = 1, 2,..., n. (1.5) (1.2), ) (1.2) (1.4) 1 (1.2) (1.4), ) (1.2) t log x i (t) log x i () = b i (u) a ij (u)x j (u) x j (u s)d s µ ij (u, s) t log x i (t) >, t > i = 1, 2,..., n x i (t) >. ] du.

9 4 1 1, 1 (1.2) (1.4) (1.2) (1.2) 2 x (t) (1.2) x(t) lim x(t) t x (t) =, R n x (t) 3 x(t) x (t) (1.2) ε >, lim x(t) t x (t) = O(e εt ), R n x (t) ε 4 (1.2) i = 1, 2,..., n ω b i(u)du >, i j a ij (t), i, j = 1, 2,..., n d s µ ij (t, s), 5 (1.2) i = 1, 2,..., n ω b i(u)du >, i j a ij (t), i, j = 1, 2,..., n d s µ ij (t, s), Lotka Volterra Lotka Volterra 6,1,11,24 26,28,3]. Lotka Volterra (), (1.2)

10 5 Lotka Volterra (permanence), (1.2) Schauder Lotka Volterra Lotka Volterra 15, 16, 27]. Lotka Volterra Lotka Volterra Lyapunov ( ) Lyapunov Lyapunov 1.3 M 4] 5 M 4]. 2 C = (c ij ) R n n, C M

11 6 (a) C (b) C T M (c) C C (d) ξ = (ξ 1, ξ 2,..., ξ n ) T Cξ (e) D CD + DC T (f) C 1 (b) (d) () 3 C = (c ij ) R n n, (a) ξ 1, ξ 2,..., ξ n n ξ jc ij > i = 1, 2,..., n (b) η 1, η 2,..., η n n η ic ji > i = 1, 2,..., n (1.2) 6 f : R + R H t > f(t), {s i, t i ]}, s i >, s i, t i ] s j, t j ] =, t i s i = t j s j > (i j), 2 i=1 ti s i f(u)du = +. a >, t > f(t) > a, f H. (1.2) 4 {f n }, ω] ω >, ω lim n f n (t) dt =, (1.6) f n, ω] ε >, k, t k, ω] n k > k f nk (t k ) ε. {f n } δ >

12 7 t k δ t t k + δ ω (1.6) f nk (t) ε 2. f nk (t) dt δ ε 2 >.

13 2.1 Schauder (1.2) f + = max{, f}, f = min{, f}, f] m = sup t f(t), f] l = inf t f(t). 1 ξ 1, ξ 2,..., ξ n, b i (t) ξ i a ii (t) ξ j a ij (t) ξ j d s µ ij (t, s)] <, t >, (2.1) ω {, j i b i (u) ξ j a + ij (u) ξ j d s µ ij (u, s)] }du + >, (2.2), j i i = 1, 2,..., n (1.2) ω C = C((, ], R n ) Banach φ = max i sup <θ φ i (θ). η ω η < min i ξ = max i ξ i, γ 1 = max i γ 2 = max i, j γ 3 = max i, j sup b i (t), t sup t sup t a ij (t), d s µ ij (t, s), K = exp{γ 1 + (γ 2 + γ 3 )ξ]ω}, L = ξγ 1 + (γ 2 + γ 3 )ξ], { b i (u) n, j i ξ ja + ij (u) n ξ j K ω a ii(u)du } d sµ ij (u, s)] + du, ξ i. (2.3)

14 9 Ω = {x(θ) C : η x i (θ) ξ i (i = 1, 2,..., n), ẋ(θ) L}, Ω C Ω C T T : φ(θ) x(θ + ω, φ), x(t) = x(t, φ) (1.2) x i (θ) = φ i (θ) ( < θ ) Schauder T Ω Ω, φ Ω, x Ω. t i x i (t ) = ξ i, x i (t) ξ i, t < t, j i x j (t) ξ j, t t. (2.1) d log x i (t) dt = b i (t ) a i j(t )x j (t ) x j (t s)d s µ i j(t, s) t=t b i (t ) ξ i a i i (t ) ξ j a i j (t ) ξ j d s µ i j(t, s)], j i <. < t ω i = 1, 2,..., n x i (t) ξ i. x i (t) η < t ω i = 1, 2,..., n < t 1 ω i 1 x i1 (t 1 ) < η, < s ω { t1 } d log x i1 (u) x i1 (t 1 s) = x i1 (t 1 ) exp du < Kη. du (2.3) t1 { d log x i1 (u) t1 du = b i1 (u) t 1 ω du t 1 ω { ω >. Kη b i1 (u) ω t 1 s a i1 j(u)x j (u), j i 1 ξ j a + i 1 j (u) a i1 i 1 (u)du x j (u s)d s µ i1 j(u, s) ξ j d s µ i1 j(u, s)] }du + } du

15 1 { t1 x i1 (t 1 ) = x i1 (t 1 ω) exp t 1 ω } d log x i1 (u) du > η. du x i1 (t 1 ) < η < t ω i = 1, 2,..., n x i (t) η. ẋ(θ + ω) L. T Ω Ω. Schauder φ Ω T φ = φ. x(t, φ ) = x(t, T φ ), x(t, φ ) = x(t + ω, φ ). x(t, φ ) (1.2) ω 3 x (t) η x i (t) ξ i, i = 1, 2,..., n. dx i (t) ] σij = x i (t) b i (t) a ij (t)x j (t) x j (t s)d s µ ij (t, s), dt i = 1, 2,..., n, (2.4) σ ij > 1 2 ξ 1, ξ 2,..., ξ n, b i (t) ξ i a ii (t) ω { b i (u), j i, j i ξ j a ij (t) ξ j d s µ ij (t, s)] <, t >, ξ j a + ij (u), j i ξ j d s µ ij (u, s)] }du + >, i = 1, 2,..., n (2.4) ω 1 < t ω i = 1, 2,..., n x i (t) η, σ = max σ ij, i, j K 1 = exp{γ 1 + (γ 2 + γ 3 )ξ](ω + σ)},

16 11 < s ω + σ { x i1 (t 1 s) = x i1 (t 1 ) exp t1 t 1 s } d log x i1 (u) du < K 1 η. du t1 d log x i1 (u) ω { σi1 du b i1 (u) ξ j a + i t 1 ω du 1 j (u) j ξ j d s µ i1 j(u, s)] }du +, j i 1, j i 1 { ω σi1 } i 1 K 1 η a i1 i 1 (u) + d s µ i1 i 1 (u, s)] ]du +. η { ω (1.2) b i (u), j i ] bj a ij (u) a jj m i = 1, 2,..., n ω ] } bj d s µ ij (u, s) du > (2.5) a jj m ( 4), 1 (2.1) (2.2) ξ i ω b i (t) ξ i a ii (t), t >, { } b i (u) ξ j a ij (u) ξ j d s µ ij (u, s) du >., j i = b i /a ii ] m (i = 1, 2,..., n), 1 2 (1.2) ζ 1, ζ 2,..., ζ n ζ i a ii (t) + ζ j a ij (t) + ζ j d s µ ij (t, s) > (2.6), j i t > i = 1, 2,..., n (1.2) ω

17 12 ( 5), 1 (2.2) ω (2.1) b i (t) ξ i a ii (t) a, j i a > max sup i t ξ j a ij (t) b i (u)du >, ξ j d s µ ij (t, s) <, t >. (2.7) b i (t) ζ i a ii (t) + n, j i ζ ja ij (t) + n ζ j d s µ ij (t, s), ξ i = aζ i (i = 1, 2,..., n), (2.7) ] Lotka Volterra dx i (t) dt = x i (t) b i (t) a ii (t)x j (t) σij b ij (t)x j (t τ ij (t)) c ij (t, s)x j (t s)ds ], i = 1, 2,..., n, (2.8) σ ij > (2.4) Lyapunov Razumikhim 1 ξ 1, ξ 2,..., ξ n α, b i (t) ξ i a ii (t) ω { σij (1 + α)ξ j b ij (t) (1 + α)ξ j b i (u), j i ξ j b + ij (t), j i c ij (t, s)ds, t >, σij ξ j c + ij }du (t, s)ds >, i = 1, 2,..., n (2.8) ω

18 13 (2.8) 2 25] 1 1 1] dx i (t) ] Tij = x i (t) b i (t) a ii (t)x j (t) a ij (t) K ij (s)x j (t s)ds, dt, j i i = 1, 2,..., n, (2.9) T ij > T ij K ij (s)ds = 1 (i j). 2 f = 1 ω ω f(u)du. a ij e y j = b i, i = 1, 2,..., n (2.1) (y 1, y 2,..., y n) T R n, b i, j i ] bj a ij > (2.11) a jj m i = 1, 2,..., n (2.9) ω 1 (2.5) (2.9) (2.11) 1, 2 (2.1) 11] dx i (t) = x i (t) b i (t) a ii (t)x i (t) + a ij (t)x j (t) dt, j i ] τj + b ij (t) x j (t s)dµ j (s), i = 1, 2,..., n, (2.12) τ j > µ j (τ j +) µ j ( ) = 1 (j = 1, 2,..., n). 3 f = 1 ω b i 2a ii e y i + ω f(u)du. (a ij + b ij )e y j =, i = 1, 2,..., n (2.13)

19 14 (y1, y2,..., yn) T R n, { } a ii ] l a ji + b ji ] m + b ii ] m > (2.14), j i i = 1, 2,..., n (2.12) ω (2.14) 3 ζ 1, ζ 2,..., ζ n { } ζ i a ii ] l ζ j a ij + b ij ] m + ζ i b ii ] m >, j i i = 1, 2,..., n 2 (2.6) 2, 3 (2.13) 2.3 (1.5) s c ij (t, s) ds < +, (2.15) τ ij (t) τ ij (t) < 1, ψ ij (t) = t τ ij (t) ψ 1 ij (t) (i, j = 1, 2,..., n). τ ij t ψ 1 ij (t) = t + τ ij (1.5) i = 1, 2,..., n ω b i(u)du >, i j a ij (t), i, j = 1, 2,..., n b ij (t) c ij (t, s), 8 (1.5) i = 1, 2,..., n ω b i(u)du >, i j a ij (t), i, j = 1, 2,..., n b ij (t) c ij (t, s), (1.5) ω 3 β i (t) = ζ i a ii (t) (1.5) ω x (t), ζ 1, ζ 2,..., ζ n, j i ζ j a ji (t) ζ j b ji (ψ 1 ji (t)) 1 τ ji (ψ 1 ji (t)) ζ j c ji (t+s, s) ds H (2.16)

20 15 i = 1, 2,..., n x (t) ε > p > log x i (t) log x i (t) p x i (t) x i (t) (2.17) ζ i (a ii (t) εp), j i ζ j a ji (t) ζ j b ji (ψ 1 ji (t)) 1 τ ji (ψ 1 ji (t)eε(ψ 1 ji (t) t) ζ j c ji (t + s, s) e εs ds (2.18) t > i = 1, 2,..., n x (t) ε. w i (t) = x i (t) x i (t), z i (t) = log x i (t) log x i (t) (i = 1, 2,..., n), L(t) = ζ i z i (t) + i=1 + ζ i i, t ζ i i, t t s t τ ij (t) b ij (ψ 1 ij (s)) 1 τ ij (ψ 1 ij (s)) w j(s) ds c ij (θ + s, s) w j (θ) dθds. (2.15) L() L(t) (1.5) dl(t) = ζ i sign(z i (t)) a ii (t)w i (t) a ij (t)w j (t) b ij (t)w j (t τ ij (t)) dt i=1, j i ] bij (ψ 1 ij c ij (t, s)w j (t s)ds + ζ (t)) i 1 τ i, ij (ψ 1 ij (t)) w j(t) ] b ij (t) w j (t τ ij (t)) + ζ i c ij (t + s, s) w j (t) ds = c ij (t, s) w j (t s) ds ζ i a ii (t) i=1, j i i, ] ζ j a ji (t) ] ζ j c ji (t + s, s) ds w i (t) β i (t) w i (t). i=1 ζ j b ji (ψ 1 ji (t)) 1 τ ji (ψ 1 ji (t))

21 16 + dl(t) dt i=1 β i (t) w i (t) dt L() < +. (2.19) L(t) x(t) (1.5) x(t), ) β i (t) H, (2.19) x (t) lim t x(t) x (t) =. i=1 L 1 (t) = ζ i z i (t) e εt + i=1 + ζ i i, i, t t s t ζ i L 1 (t) (1.5) t τ ij (t) b ij (ψ 1 ij (s)) 1 τ ij (ψ 1 ij (s)) w j(s) e εψ 1 ij (s) ds c ij (θ + s, s) w j (θ) e ε(θ+s) dθds. dl 1 (t) dt = ζ i sign(z i (t))e εz εt i (t) a ii (t)w i (t) i=1 + + b ij (t)w j (t τ ij (t)) e εt. i,, j i a ij (t)w j (t) c ij (t, s)w j (t s)ds bij (ψ 1 ζ i e εt ij (t)) ] 1 τ ij (ψ 1 ij (t)) w j(t) e ε(ψ 1 ij (t) t) b ij (t) w j (t τ ij (t)) ] ζ i e εt c ij (t + s, s) w j (t) e εs ds c ij (t, s) w j (t s) ds i, ζ i (a ii (t) εp) i=1, j i ] ζ j c ji (t + s, s) e εs ds w i (t) ζ j a ji (t) ζ j ] b ji (ψ 1 ji (t)) 1 τ ji (ψ 1 ji (t))eε(ψ 1 ji (t) t) L 1 (t) L 1 (t) ζ i log x i (t) log x i (t) L 1 ()e εt. (2.2) i=1

22 17 x (t) m 1 m 2 m 1 x i (t) m 2 t > i = 1, 2,..., n q x i (t) x i (t) q log x i (t) log x i (t) (2.21) t > i = 1, 2,..., n (2.2) (2.21) ζ i x i (t) x i (t) ql 1 ()e εt. (2.22) i=1 x {ζ,1} = (2.22) ζ i x i, i=1 x(t) x (t) {ζ,1} = O(e εt ). x (t) ε. 4 x (t) m 1 m 2 m 1 x i (t) m 2 t > i = 1, 2,..., n (2.17) p 3, 25, 26] (2.16) ξ 1, ξ 2,..., ξ n, ζ 1, ζ 2,..., ζ n b i (t) ξ i a ii (t) ξ j a ij (t) ξ j b ij (t) ξ j c ij (t, s)ds <, t >, ω ζ i a ii (t), j i b i (u), j i, j i ζ j a ji (t) ξ j a + ij (u) ξ j b + ij (u) ξ j c + ij ]du (u, s) >, ζ j b ji (ψ 1 ji (t)) 1 τ ji (ψ 1 ji (t)) ζ j c ji (t + s, s) ds H i = 1, 2,..., n (1.5) ω x (t). ε > p > log x i (t) log x i (t) p x i (t) x i (t)

23 18 ζ i (a ii (t) εp), j i ζ j a ji (t) ζ j b ji (ψ 1 ji (t)) 1 τ ji (ψ 1 ji (t)eε(ψ 1 ji (t) t) ζ j c ji (t + s, s) e εs ds t > i = 1, 2,..., n x (t) ε. { ω 3 b i (u) ζ i a ii (t) (1.5) ζ 1, ζ 2,..., ζ n, j i, j i ] bj a ij (u) a jj m ζ j a ji (t) ζ j ] bj b ij (u) a jj m b ji (ψ 1 ji (t)) 1 τ ji (ψ 1 ji (t)) ] } bj c ij (u, s)ds du >, a jj m ζ j c ji (t + s, s) ds H i = 1, 2,..., n (1.5) ω x (t). ε > p > log x i (t) log x i (t) p x i (t) x i (t) ζ i (a ii (t) εp), j i ζ j a ji (t) ζ j b ji (ψ 1 ji (t)) 1 τ ji (ψ 1 ji (t)eε(ψ 1 ji (t) t) ζ j c ji (t + s, s) e εs ds t > i = 1, 2,..., n x (t) ε ξ i a ii (t) + ζ i a ii (t) (1.5) ξ 1, ξ 2,..., ξ n, ζ 1, ζ 2,..., ζ n, j i, j i ξ j a ij (t) + ζ j a ji (t) ξ j b ij (t) + ζ j ξ j c ij (t, s)ds >, t >, b ji (ψ 1 ji (t)) 1 τ ji (ψ 1 ji (t)) ζ j c ji (t + s, s) ds H

24 19 i = 1, 2,..., n (1.5) ω x (t). ε > p > ζ i (a ii (t) εp) log x i (t) log x i (t) p x i (t) x i (t), j i ζ j a ji (t) ζ j b ji (ψ 1 ji (t)) 1 τ ji (ψ 1 ji (t)eε(ψ 1 ji (t) t) ζ j c ji (t + s, s) e εs ds t > i = 1, 2,..., n x (t) ε. 2 3

25 3.1 Lotka Volterra (1.2) 5 ξ 1, ξ 2,..., ξ n ξ i a ii (t) + ξ j a ij (t) + ξ j d s µ ij (t, s)] > (3.1), j i t < ω i = 1, 2,..., n (1.2) x(t) M i x i (t) M i t > i = 1, 2,..., n x(t) α i = inf t { ξ i a ii (t) +, j i H = max i { λ > max i } ξ j a ij (t) + ξ j d s µ ij (t, s)] >, { } sups φ i (s), sup t<ω ξ i b + i (t) α i M i = λξ i (i = 1, 2,..., n). φ i (s) Hξ i λξ i = M i s t i ], H x i (t ) = M i, x i (t) M i, t < t, }, j i x j (t) M j, t t.

26 21 d log x i (t) dt = b i (t ) a i j(t )x j (t ) x j (t s)d s µ i j(t, s) t=t b i (t ) a i i (t )M i a i j (t )M j M j d s µ i j(t, s)], j i { } = b i (t ) λ ξ i a i i (t ) + ξ j a i j (t ) + ξ j d s µ i j(t, s)], j i b i (t ) λα i <, t > i = 1, 2,..., n x i (t) M i. x(t) (1.2) x(t) x(t) 5 x(t) M i H, x(t) 31] 3.2 (1.2) (1.2) ω 6 (1.2) η 1, η 2,..., η n η i a ii (t) η j a ji (t) η j dk ji (s) > (3.2), j i t < ω i = 1, 2,..., n (1.2) ω w i (t) = x i (t+ω) x i (t), z i (t) = log x i (t+ω) log x i (t) (i = 1, 2,..., n), L(t) = η i z i (t) + η i t i=1 i, t s w j (θ) dθ dk ij (s).

27 22 (2.15) L() { } β = min inf η i a ii (t) η j a ji (t) η j dk ji (s) >. i t, j i L(t) (1.2) dl(t) = η i sign(z i (t)) a ii (t)w i (t) dt i=1 + η i w j (t) dk ij (s) i, η i a ii (t) w i (t) + i=1 + =, j i η i w j (t) dk ij (s) i, η i a ii (t) i=1 β w(t),, j i η j a ji (t), j i a ij (t)w j (t) w j (t s) dk ij (s) a ij (t) w j (t) + w j (t s) dk ij (s) ] w j (t s)d s µ ij (t, s) w j (t s) dk ij (s) ] ] η j dk ji (s) w i (t) w(t) = n i=1 w i(t). + L(t) w(t) n=1 ω w(t) dt 1 L() < +. β x(t + nω) x(t + (n 1)ω) dt < +. Cauchy x(t + nω) L 1, ω] x(t) (1.2) x(t) {x(t + nω)} Arzéla Ascoli {x(t + n k ω)} R x (t). x (t) {x(t + nω)} L 1 ω lim x(t + nω) x (t) dt =. n f n (t) = x(t + nω) x (t). 4 f n (t), ω] x(t + nω) x (t), ω] x(t + nω) x (t) R ] ]

28 23 x (t) (1.2) ω x (t + ω) = lim n x ( t + (n + 1)ω ) = lim n x(t + nω) = x (t), x (t) ω (1.2) x nk (t) x(t), x nk (t) i x n k i dx n k i (t) dt = x n k i (t) b i (t) a ij (t)x n k j (t), (t) x n k j (t s)d sµ ij (t, s) ] i = 1, 2,..., n. k (1.3) {x nk (t)} R x i (t) b i (t) dx i (t) = x i (t) b i (t) dt a ij (t)x j(t) a ij (t)x j(t) x (t) (1.2). t = t + nω, t < ω, x j(t s)d s µ ij (t, s) x j(t s)d s µ ij (t, s) x(t) x (t) = x(t + nω) x (t ). {x(t + nω)}, ω] ] ]., i = 1, 2,..., n, lim x(t) t x (t) =. (3.3) (1.2) x (t). y(t) w i (t) = y i (t) x i (t) z i (t) = log y i (t) log x i (t) (i = 1, 2,..., n). x(t) y(t) (3.3) y(t) x(t) dt < +. lim y(t) x(t) =. t, lim y(t) t x (t) =.

29 24 ( 5) (3.1) (3.2), M a ii = inf t a ii (t) a ij = sup t a ij (t) (i j). 3 7 ξ 1, ξ 2,..., ξ n ξ i a ii ξ j a ij ξ j dk ij (s) > (3.4), j i i = 1, 2,..., n (1.2) ω (3.4) (3.1) 5 (1.2) a ii C = (c ij ) = a ij dk ii (s), i = j,, j i dk ij (s), i j. 3 (3.4) η 1, η 2,..., η n η i a ii η j a ji η j dk ji (s) > (3.5) i = 1, 2,..., n (3.2) 6 ξ 1, ξ 2,..., ξ n (3.4) ξ i = 1 (i = 1, 2,..., n), (3.4) ( 1). 7 2, 5 (3.5) C M (1.2) ω M 2 (a), M

30 25 6 Gauss 2] 3 3 2]

31 4.1 (1.2) 1 Lotka Volterra dx 1 (t) = x 1 (t)7 + sin t (6 + sin t)x 1 (t) (2 + cos t)x 2 (t 1) + (1 + sin t)x 3 (t 1)], dt dx 2 (t) = x 2 (t)4 + cos t (1 + sin t)x 1 (t 1) (5 + sin t)x 2 (t) + (2 + cos t)x 3 (t 1)], dt dx 3 (t) = x 3 (t)1 + sin t (3 + cos t)x 1 (t 1) + (1 + cos t)x 2 (t 1) (9 + cos t)x 3 (t)]. dt (3.5) C = C (3.4) ξ i = 1 (i = 1, 2,..., n) >, = 14 >, = 6 > (a) C M 5 1 ω Matlab dde t 2π 4.3

32 x i t π x(s) (1,.5,.5) T.

33 x x x x 3 = 2π x(s) (1,.5,.5) T, x(s) (.5, 1,.5) T x(s) (.5,.5, 1) T. x 3 = 1 (b 1 (t), b 2 (t), b 3 (t)) (7+sin t, 4+cos t, 4+sin t), (7+sin t, 1+cos t, 1+sin t) ( 1+sin t, 1+ cos t, 1 + sin t), (1.2) ω , 8 (1.2) ξ 1, ξ 2,..., ξ n ξ i a ii ξ j a ij ξ j dk ij (s) > (4.1), j i i = 1, 2,..., n ω

34 29 x i x i (a) (b) (c) t t.8.6 x i t (b 1 (t), b 2 (t), b 3 (t)) (a) (7 + sin t, 4 + cos t, 4 + sin t), (b) (7 + sin t, 1 + cos t, 1 + sin t), (c) ( 1 + sin t, 1 + cos t, 1 + sin t).

35 3 (4.1) 2 (2.6). 7 2 (1.2) ω x (t), ω x +(t). lim t x +(t) x (t) =. x (t) x +(t). ω (1.2) 1 Lotka Volterra (1.2) dk ij (s). Lotka Volterra dx i (t) = x i (t) b i (t) dt a ij (t)x j (t) ] b ij (t)x j (t τ ij ), i = 1, 2,..., n. 7, ξ 1, ξ 2,..., ξ n ξ i a ii, j i i = 1, 2,..., n a ii ξ j a ij ξ j b ij > = inf t a ii (t), a ij = sup t a ij (t) (i j), b ij = sup t b ij (t). τ ij

36 31 (3.1) (3.2) t, 7 (3.4) (3.1) (3.2) 3 1 ξ 1, ξ 2,..., ξ n ξ i a ii (t) ξ j a ij (t) ξ j dk ij (s) >, j i t < ω i = 1, 2,..., n (1.2) ω Lotka Volterra (1) (2) (3) 31] Lotka Volterra (complete stability), () Lotka Volterra Lotka Volterra

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