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‘k;J¤?? ???? ? ? X? ????? _??? ì? - 041004) ( ì?“‰??ê??O??‰?? ? ? ??????a ?k;J¤,… ?m?p? ,?k HollingII .?U?A?ê ? ?.. ?L???§ ? 5?, X?±?5 ?? ^?.????L E?A o??ì??ê ?{ ±?) ?ìC-?5. '…c ? X? ;J¤ ±?5 ±?) ?ìC-? ???a?:O175.1 MR ?a?:34D20;92D25 ?zI?è:A §1 ?ó CAc'u? ?X? ????ké?. ?[6]3‘k Holling II .?U?A ?ê ü? ??? ? . ?:?,?? ‘k;J¤ Holling II .? ?? ? ? .,…?? 1???1 ? ?+? ???? ] ? ?^.X ??±e/?: ? β1 (1?m1 )x1 x3 ? 1 ? x = α1 x1 (1 ? x ? k1 ) ? σx1 x2 ? 1+a1 (1?m1 )x1 ? 1 β2 (1?m2 )x2 x3 2 x2 = α2 x2 (1 ? x k2 ) ? σx2 x2 ? 1+a2 (1?m2 )x2 ? ? ? ? x = ?γx3 + c1 β1 (1?m1 )x1 x3 + c2 β2 (1?m2 )x2 x3 3 1+a1 (1?m1 )x1 1+a2 (1?m2 )x2 (1.1) ??^? 0 0 x1 (0) = x0 1 >0, x2 (0) = x2 >
0, x3 (0) = x3 >
0, d? x1 , x2 L??? t ?? ?+ —?, x3 L?? ??+3?? t ?? —?,b X? (1.1) ¤k ?ê?? ~ê, α1 , α2 L??A 1???1 ? ?+ O ??, γ L?? ? k ?, k1 , k2 L? ?+ ??NB?, β1 , β2 L?? ?+é β1 β2 1???1 ? ?+ |??, a1 , a2 L?ü ?mS1???1 ? ?+ ? 1 1 ?? ???, c1 , c2 ?=??f?,?A/ a1 , a2 ?§? ? ??,L?z?? β 1 x1 β 2 x2 #) ? ???1 ???1 ? ?+ ê?,… 0 <
c1 , c1 <
1 . 1+ a1 x1 , 1+a2 x2 ùü‘ L?? ?é ?+ ?^,=?U?A?ê? Holling II .. m1 x1 ? m2 x2 L?‘k;
J¤ —?,?ü? —??¤ '
,… 0 <
m1 , m2 <
1 , (1 ? m1 )x1 ? (1 ? m2 )x2 ì??g,‰??7]?(2013011002-2) ??? (1986–), ?, ì?“‰??ê??O??‰?? 3???). ? ????. _?? (1963—) I, ì?$?<, ì?“‰??ê??O??‰?? a?, ??l?)?ê? ???? E-mail : jiajw.2008@163.com ? ? ?, 1 ??U ? ?? ü? ?+, σ1 , σ2 ?1?? 1 ? ?? O??. σ1 , σ2 ???'?Xê,d?*?. m ? ü$ ? + O ? ?. · ? 3 . (1.1) ?:?‰UC,=???g? X?,?…O\ rc ?¤c ??,? ? —?? . .Xe? ? β1 (t)(1?q1 (t))x1 x3 ? ? x (t) = x1 (a1 (t) ? b1 (t)x1 ) ? σ1 (t)x1 x2 ? 1+ ? α1 (t)(1?q1 (t))x1 ? h1 (t)x1 ? 1 β2 (t)(1?q2 (t))x2 x3 x2 (t) = x2 (a2 (t) ? b2 (t)x2 ) ? σ2 (t)x1 x2 ? 1+ α2 (t)(1?q2 (t))x2 ? h2 (t)x2 ? ? ? ? x (t) = ?γ3 (t)x3 + c1 (t)β1 (t)(1?q1 (t))x1 x3 + c2 (t)β2 (t)(1?q2 (t))x2 x3 ? h3 (t)x3 ? ρ(t)x2 3 1+α1 (t)(1?q1 (t))x1 1+α2 (t)(1?q2 (t))x2 3 (1.2) ? . (1.1) ,?? x1 (t), x2 (t) L? ?+ —?, ? ?d σ1 (t) ? σ2 (t) 5?? .b —?, x3 (t) L?3? ?3 t ?? ?+ (H1 ) : ai (t) , bi (t) , σi (t) , αi (t) , βi (t) , qi (t) , ci (t) (i = 1, 2) , γ3 (t) , ρ(t) , hj (t) (j = 1, 2, 3) 3 R ???±?? ω ?Y ±??ê,…÷v l l l l l l l l l min{ al i , bi , σi , αi , βi , qi , ci , γ3 , ρ , hj } >
0, u u u u u u u u u max{ au i , bi , σi , αi , βi , qi , ci , γ3 , ρ , hj } <
+∞, (i = 1, 2;
j = 1, 2, 3). §2 ù??·?0 ? ? ????n ? ?n. ???3 ????^ 3 = {(x , x , x ) ∈ R3 | x - R+ 1 2 3 1,2,3 >
0} ,PX? (1.2) ^? ) (x1 (t), x2 (t), x3 (t)) ÷v?? 0 0 (x0 1 , x2 , x3 ) = (x1 (t0 ), x2 (t0 ), x3 (t0 )), t0 ≥ 0. + S . ??)????, ???·??I??k ??? ),=3 R3 - g (t) ??Y?ê,? {B·?{ ? g .e g 3 R ??k. ,·?P g l = inf g (t), 1 ω ω 0 g u = sup g (t), t ∈ R. e g ?±?? ω ±??ê, ·?P g ?= g (t)dt . ?? 2.1 e?3 ~ê Mi , m1 , m2 , m3 ? X? (1.2) t→∞ t→∞ ??) (x1 , x2 , x3 )?k mi ≤ lim inf xi (t) ≤ lim sup xi (t) ≤ Mi , (i = 1, 2, 3). K?X? (1.2) ?±? . ?? 2.2 eX? (1.2) ?k ??? ??) x(t) = (x1 , x2 , x3 ) ,÷v 3 t→∞ lim | xi (t) ? x? i (t) |= 0, i=1 KX? (1.2) k. ?K) x? (t) ? ?ìC-? . 2 ?? 2.3 eéX? (1.2) k??? ??),?3 T1 >
0 ,÷vé¤k 3 ,K8? A ?X? (1.2) x(t) ∈ A ? R+ ??k.??. + ?n 2.1 R3 ?X? t ≥ t0 + T1 ,k ??C8. X? (1.2) ) (x1 , x2 , x3 )÷v 0 0 y? ‘??^? (x0 1 , x2 , x3 ) ? ? ? x = x0 ? 1 exp{ ? 1 x2 = x0 2 exp{ ? ? ? ? x3 = x0 exp{ 3 t t0 [a1 (t) ? t t0 [a2 (t) ? t t0 [γ3 (t) + b1 (t)x1 ? σ1 (t)x2 ? β1 (t)(1?q1 (t))x3 1+α1 (t)(1?q1 (t))x1 ? h1 (t)]du} β2 (t)(1?q2 (t))x3 b2 (t)x2 ? σ2 (t)x1 ? 1+ α2 (t)(1?q2 (t))x2 ? h2 (t)]du} c1 (t)β1 (t)(1?q1 (t))x1 c2 (t)β2 (t)(1?q2 (t))x2 1+α1 (t)(1?q1 (t))x1 + 1+α2 (t)(1?q2 (t))x2 ? h3 (t) ? ρ(t)x3 ]du} é¤k t ∈ [t0 , ∞) ¤á.y.. ?n 2.2 ([6]) h ????ê, f ?????3 [h, +∞) ? ?K?ê,…k f 3 [h, +∞) ???? ??—?Y ,K limt→∞ f (t) = 0 . §3 ?n 3.1 b X? (1.2) ÷v (H1 ) 9 ±?5 u u l u (H2 ) al 1 ? σ1 M2 ? β1 (1 ? q1 )M3 ? h1 >
0 u u l u (H3 ) al 2 ? σ2 M1 ? β2 (1 ? q2 )M3 ? h2 >
0 u+ (H4 ) ? r3 l u l u cl cl u 1 β1 (1?q1 )m1 2 β2 (1?q2 )m2 + u l u l )M ) ? h3 >
0 1+a1 (1?q1 )M1 1+a2 (1?q2 2 {(x1 , x2 , x3 ) ∈ R3 | mi <
xi <
Mi } ?X? (1.2) K Γ0 = ? .?? ??C8§ =X? (1.2) ?± m1 = u u l u al 1 ? σ1 M2 ? β1 (1 ? q1 )M3 ? h1 , bu 1 m2 = u u l u al 2 ? σ2 M1 ? β2 (1 ? q2 )M3 ? h2 , bu 2 ?ru m3 = u3 + ρ M1 = au 1 , bl 1 l u cl 1 β1 (1?q1 )m1 l 1+au (1 ? q 1 1 )M3 u ρ + l u cl 2 β2 (1?q2 )m2 ) l 1+au (1 ? q 2 2 )M3 ρu ? hu 3 . ρu (3.1) M2 = au 2 , bl 2 M3 = u u u l l cu 1 β1 + c2 β2 ? r3 ? h3 l ρ y? dX? (1.2) 1???§ , l x1 ≤ x1 (au 1 ? b1 x1 ) 0 ≤ x1 (0) ≤ M1 = au 1 bl 1 ?, 0 ≤ x1 (t) ≤ M1 (t >
0). XJ x1 (0) >
M1 ,K7?3?? T1 >
0 , t ≥ T1 ?, x1 (t) ≤ M1 .?Ké¤k t ≥ 0, x1 (t) >
M1 ,K?3??~ê η1 >
0 ,? x1 (t) ? M1 ≥ η1 ,dX? (1.2) ? l l x1 ≤ x1 [au 1 ? b1 (M1 + η1 )] = ?b1 η1 x1 Kk x1 ≤ x1 (0) exp(?bl 1 η1 t) . g?.¤±?3?? T1 >
0, t → +∞ ?,??mà?u0,ù?é¤k t ≥ T1 ?,·?k x1 (t) ≤ M1 . 3 t ≥ 0, x1 (t) >
M1 dX? (1.2) 1 ??§? l x2 ≤ x2 (au 2 ? b2 x2 ) au 2 bl 2 0 ≤ x2 (0) ≤ M2 = ?, 0 ≤ x2 (t) ≤ M2 , t >
0. t ≥ T2 ?, x2 (t) ≤ M2 . XJ x2 (0) >
M2 ,?n,?3?? T2 >
0, dX? (1.2) 1n??§? l l u u u u x3 ≤ x3 (?r3 ? hl 3 ? ρ x3 + c1 β1 + c2 β2 ). u u u l l cu 1 β1 +c2 β2 ?r3 ?h3 ρl K 0 ≤ x3 (0) ≤ M3 = ?, x3 (t) ≤ M3 . t ≥ T3 ?, x3 (t) ≤ M3 . XJ x3 (0) >
M3 ,?n,?3?? T3 >
0 , dX? (1.2) 1???§ , u u u l u u . x1 ≥ x1 [al 1 ? b1 x1 ? σ1 M2 ? β1 (1 ? q1 )M3 ? h1 ] = x1 b1 (?x1 + m1 ) (3.2) x1 (0) ≥ m1 ,Kk x1 (t) ≥ m1 . XJ 0 <
x1 (0) <
m1 ,K7?3?? T4 >
0 , t ≥ T4 ?, x1 (t) ≥ m1 . ?Ké¤k t ≥ 0 ,k x1 (t) <
m1 .K?3?? ~ê η2 >
0 ,? x1 (t) ? m1 ≤ ?η2 ,dX? (1.2) u u u l u u . x1 ≥ x1 [al 1 ? b1 (m1 ? η2 ) ? σ1 M2 ? β1 (1 ? q1 )M3 ? h1 ] = b1 η2 x1 Kk x1 (t) ≥ x1 (0) exp(bu t → +∞ ?,???u +∞ .ù?é¤k 1 η1 t) , g?.¤±?3?? T4 >
0 , t ≥ T4 ?, x1 (t) ≥ m1 . ?q/§ dX? (1.2) 1 ??§??: t ≥ 0, x1 (t) <
m1 u u u l u u x2 ≥ x2 [al 2 ? b2 x2 ? σ2 M1 ? β2 (1 ? q2 )M3 ? h2 ] = x2 b2 (?x2 + m2 ). (3.3) K?3?? T5 >
0 , dX? (1.2) t ≥ T5 ?, x2 (t) ≥ m2 . 1n??§? : + l u cl 2 β2 (1?q2 )m2 ) l 1+au (1 ? q 2 2 )M2 u+ x3 ≥ x3 [?r3 l u cl 1 β1 (1?q1 )m1 l 1+au (1 ? q 1 1 )M1 u ? hu 3 ? ρ x3 ] = ρu x 3 (?x3 + u ?r3 ρu + l u cl 1 β1 (1?q1 )m1 l 1+au (1 ? q 1 1 )M1 ρu + l u cl 2 β2 (1?q2 )m2 ) l 1+au (1 ? q 2 2 )M2 ρu ? hu 3 ρu ) (3.4) = ρu x3 (?x3 + m3 ) ?n? ?3?? T6 >
0 , n?, t ≥ T6 ?, x3 (t) ≥ m3 . . t ≥ T = max{T1 , T2 , T3 , T4 , T5 , T6 } ?,X??±? 5? d?n ^?? ;J¤ q1 , q2 é±?5k?? K?. 4 §4 ?n 4.1 3?n3.1 ±?) ?ìC-?5 ?3???? ω ±?). ^?e ,X?– y? d (H1 ) ?,X? ¤kXê?? ω ±??ê,KX? (1.2) ???±?X?.P 0 0 X? (1.2) ÷v??^? X0 = (x0 )? X (t, X0 ) = (x1 (t, X0 ), x2 (t, X0 ), x3 (t, X0 )) 1 , x2 , x3 ) + + + ???? P oincare N A : R3 → R3 . A(X0 ) = X (ω, x0 ), x ∈ R3 . ?d,X? (1.2) ±? + ) ?35 duN A ??: ?35. Γ0 ? R3 k.4à8,…? ??C8, X0 ∈ Γ0 , X (t, X0 ) ∈ Γ0 ,= AΓ0 ? Γ0 ,d)é?? ?Y5? A ??Y . d Brouwer ? ?:?n?, A 3 Γ0 ?– k????:,?dX? (1.2) – ?3???? ω ±? ). ?n 4.2 3?n3.1 t→∞ ^?e ,?…b (H5 ) : lim sup[?b1 ? lim sup[?b2 ? t→∞ a1 β1 (1 ? q1 )2 m3 + c1 β1 (1 ? q1 ) + σ2 ] <
0, (1 + a1 (1 ? q1 )m1 ) a2 β2 (1 ? q2 )2 m3 + c2 β2 (1 ? q2 ) + σ1 ] <
0, (1 + a2 (1 ? q2 )m2 ) (3.6) t→∞ lim sup[?ρ + β1 (1 ? q1 ) + β2 (1 ? q2 )] <
0. KX? (1.2) ?3 ?ìC-? ±?). y? x? ?? ω ±?), xi ?X? (1.2) ?k ?? ??), d i ?X? (1.2) ?n 3.1 ? Γ0 ?X? (1.2) ??k.??.K?3 T1 >
0 , t ≥ t0 + T1 ?, xi , x? i ∈ Γ0 . Eo??ì??ê 3 V (t) = i=1 | ln xi ? ln x? i |, t ≥ t0 . ÷ (1.2) D+ V = )? ? V (t) 3 i=1 sign(xi m x ê D+ V , x? i x? i i ? x? i )( xi ? ) x? 3 )] 1+a1 (1?q1 )x? 1 ? x3 x3 q2 )( 1+a2 (1 )] ?q2 )x2 ? 1+a2 (1?q2 )x? 2 x3 ? ? = sign(x1 ? x? 1 )[?b1 (x1 ? x1 ) ? σ1 (x2 ? x2 ) ? β1 (1 ? q1 )( 1+a1 (1?q1 )x1 ? ? ? +sign(x2 ? x? 2 )[?b2 (x2 ? x2 ) ? σ2 (x1 ? x1 ) ? β2 (1 ? x1 +sign(x3 ? x? 3 )[c1 β1 (1 ? q1 )( 1+a1 (1?q1 )x1 ? x2 +c2 β2 (1 ? q2 )( 1+a2 (1 ?q2 )x2 ? x? 2 ) 1+a2 (1?q2 )x? 2 x? 1 ) 1+a1 (1?q1 )x? 1 ? ρ(x3 ? x? 3 )] d???ê 5?9?? ? ‘ (x ?x? )+a (1?q )(x? x ?x x? ) 1 ? ? (x3 ?x? 3 )+a2 (1?q2 )(x2 x3 ?x2 x3 ) x? 2 ) (1+a2 (1?q2 )x2 )(1+a2 (1?q2 )x? 2) 3 1 1 1 3 ? ? 3 1 3 D+ V ≤ ?b1 | x1 ? x? 1 | +σ1 | x2 ? x2 | ?β1 (1 ? q1 )sign(x1 ? x1 ) (1+a1 (1?q1 )x1 )(1+a1 (1?q1 )x? ) ?b2 | x2 ? x? 2 | +σ2 | x1 ? x? 1 | ?β2 (1 ? q2 )sign(x2 ? x ? x? 1 1 1 +c1 β1 (1 ? q1 )sign(x3 ? x? 3 ) (1+a1 (1?q1 )x1 )(1+a1 (1?q1 )x? ) +c2 β2 (1 ? q2 )sign(x3 ? xi , x? i ∈ Γ0 ,é t ≥ t0 + T1 ,5? x2 ? x? 2 x? 3 ) (1+a2 (1?q2 )x2 )(1+a2 (1?q2 )x? 2) ? ρ | x3 ? x? 3 | ? ? ? x3 x? 1 ? x3 x1 = x3 (x1 ? x1 ) + x1 (x3 ? x3 ) 5 ? ? ? x3 x? 2 ? x3 x2 = x3 (x2 ? x2 ) + x2 (x3 ? x3 ) ? D+ V ≤ ?b1 | x1 ? x? 1 | +σ1 | x2 ? x2 | ?β1 (1 ? q1 )sign(x1 ? x? 1) ?β2 (1 ? q2 )sign(x2 ? x? 2) +c1 β1 (1 ? q1 )sign(x3 ? +c2 β2 (1 ? q2 )sign(x3 ? ? (1+a1 (1?q1 )x1 )(x3 ?x? 3 )+a1 (1?q1 )x3 (x1 ?x1 ) (1+a1 (1?q1 )x1 )(1+a1 (1?q1 )x? ) 1 ? ?b2 | x2 ? x? 2 | +σ2 | x1 ? x1 | ? (1+a2 (1?q2 )x2 )(x3 ?x? 3 )+a2 (1?q2 )x3 (x2 ?x2 ) ) (1+a2 (1?q2 )x2 )(1+a2 (1?q2 )x? 2 x1 ? x? 1 x? ) ? 3 (1+a1 (1?q1 )x1 )(1+a1 (1?q1 )x1 ) x2 ?x? 2 x? ? ρ | x3 ? 3 ) (1+a2 (1?q2 )x2 )(1+a2 (1?q2 )x? 2) x? 3 | ? D+ V ≤ ?b1 | x1 ? x? 1 | +σ1 | x2 ? x2 | a1 β1 (1?q1 ) x3 1 (1?q1 )(1+a1 (1?q1 )x1 ) ? ? + (1+β a1 (1?q1 )x1 )(1+a1 (1?q1 )x? ) | x3 ? x3 | ? (1+a1 (1?q1 )x1 )(1+a1 (1?q1 )x? ) | x1 ? x1 | 1 1 2 ? ?b2 | x2 ? x? 2 | +σ2 | x1 ? x1 | a2 β2 (1?q2 ) x3 2 (1?q2 )(1+a2 (1?q2 )x2 ) ? ? + (1+β a2 (1?q2 )x2 )(1+a2 (1?q2 )x? ) | x3 ? x3 | ? (1+a2 (1?q2 )x2 )(1+a2 (1?q2 )x? ) | x2 ? x2 | 2 1 2 2 2 c2 β2 (1?q2 ) 1 β1 (1?q1 ) ? ? + (1+a1 (1?qc ? | x1 ? x1 | + (1+a (1?q )x )(1+a (1?q )x? ) | x2 ? x2 | 1 )x1 )(1+a1 (1?q1 )x ) 2 2 2 2 2 ?ρ | x3 ? x? 3 | = [?b1 ? a1 β1 (1?q1 )2 x3 +c1 β1 (1?q1 ) + σ2 ] | x1 ? x? 1 | (1+a1 (1?q1 )x1 )(1+a1 (1?q1 )x? 1) 2 a2 β2 (1?q2 ) x3 +c2 β2 (1?q2 ) +[?b2 ? (1+ + σ1 ] | x2 ? x? 2 a2 (1?q2 )x2 )(1+a2 (1?q2 )x? 2) β2 (1?q2 ) β1 (1?q1 ) ? +[?ρ + 1+a1 (1?q1 )x? + 1+a2 (1?q2 )x? ] | x3 ? x3 | 1 2 ?q1 )2 m3 +c1 β1 (1?q1 ) ? [?b1 ? a1 β1 (1 + σ 2 ] | x1 ? x1 | (1+a1 (1?q1 )m1 ) ?q2 )2 m3 +c2 β2 (1?q2 ) +[?b2 ? a2 β2 (1 + σ1 ] | x2 ? x? 2 | (1+a2 (1?q2 )m2 ) | ≤ +[?ρ + β1 (1 ? q1 ) + β2 (1 ? q2 )] | x3 ? x? 3 |, t ≥ t0 + T1 . (? (3.6) ????? ?3 ? >
0 , T2 ≥ t0 + T1 ÷v 3 D V ≤ ?? i=1 + | xi ? x? i |, t ≥ T2 . (3.7) é (3.7) ? ü>l T2 t ?? t 3 V (t) + ? ( T2 i=1 | xi ? x? i |)ds ≤ V (T2 ) <
∞, t ≥ T2 . 3 i=1 1 | xi ? x? i |∈ L ([T2 , ∞)) . K t T2 ( 3 i=1 ?1 | xi ? x? i |)ds ≤ ? V (T2 ) <
∞, t ≥ T2 , ?d ,???,dé¤k t ≥ t0 + T1 ,k x, x? ∈ Γ0 ??§ (1.2) ?? xi (t), x? ê i (t) 3 ? 3 [T2 , ∞) ?k. .?d i=1 | xi ? xi | 3 [T2 , ∞) ???—?Y . d?n (2.2) ? ? limt→∞ 3 i=1 | xi ? xi |= 0 ,y.. ???z [1] I.Barblat,Systemes dequation di?erentielles dosillation non linearires, Rev.Romaine Math.Pures Appl.4(1975)267-270. 6 [2] D.D.Bainov,P.S.Simeonov,Impulsive Di?erent Equations:Periodic Solution and Application, Pitman Momographs and Surveys in Pure and Applied Mathematics,(1993) [3] Y.J.Huang,F.D.Chen,L.Zhong,Stability analysis of a prey-predator model with Holling type III response function incorporating a prey refuge.Applied Mathematics and Computation,182(2006)672683. [4] T.K.Kar,Stability analysis of a prey-predator model incorporation a prey refuge, Commun.Nonlinear Sci.Numer.Simul.10(2005)681-691. [5] E.G.Olivares,R.R.Jiliberto,Dynamics consequents of prey refuge in a simple model system: more preys and few peredators and enhanced stability.Ecol.Model.166(2003)135-146. [6] S.Sarwardi,P.K.Mandal,S.Ray,Analysis of a competitive prey-predator system with a prey refuge, Biosystems 110(2012)133-148. [7] T.V,Ton,Dynamic of species in a Acad.Paedagog.Nyhazi.25(2009)45-54 non-autonomous Lotka-Volterra system,Acta Math. [8] T.V.Ton,N.T.Hieu,Dynamics of species in a model with two predators and one prey, Nonlinear Analysis 74(2011)4868-4881. Dynamic behaviors of a non-autonomous predator-prey model incorporating a prey refuge Yamin Hou Jianwen Jia ( School of Mathematical and Computer Science , Shanxi Normal University, Shanxi , Linfen , 041004,P.R. China) Abstract In this paper,we study a competitive predator-prey model which has one predator and two preys with Holling type-II functional response incorporating a constant proportion prey refuge. And using ordinary di?erential equation qualitative and stability method, we get the persistence of the system under certain conditions.Further by constructing lyapunov function method, we get the global asymptotic stability of the periodic solution. Keywords Predator-prey model Refuge;
Persistence;
Periodic solution;
Global asymptotic stability 7
第 7 期 ( 2000 年 7 月 ) Applied Mathemat ics and Mechanics 文章编号 : 1000_0887( 2000) 07_0693_08 应用数学和力学编委会编 重 庆 出 版 社 出 版 一类微生物种群生态数学...
* 9xm, http:// 单位:段美元 王寿松 * (中山医科大学数学教研室 广州510089) 9xm, http:// 关键词: 营养 上升函数;赤潮;单种群;生态数学模型...
伊犁师范学院学报;2003年02期5滕志东;段魁臣;; 三种群生态系统的持续生存 [J];生物数学学报;1990年01期6陆志奇;; 二个替换资源的开发竞争(Ⅱ) [J];河南师范大学...