Global Solutions of the SKT Model in Population Dynamics

Size: px

Start display at page:

Download "Global Solutions of the SKT Model in Population Dynamics"

Loraine Manning
5 years ago
Views:

Volm 7 No 7 499-5 ISSN: 3-88 rin rion; ISSN: 34-3395 on-lin rion rl: h://ijm ijm Glol Solion of h SK Mol in Polion Dnmi Rizg Hor n Mo Soilh USH El li Ezzor lgir lgri rizg@gmilom USH El li Ezzor lgir

1 Volm 7 No ISSN: 3-88 rin rion; ISSN: on-lin rion rl: h://ijm ijm Glol Solion of h SK Mol in Polion Dnmi Rizg Hor n Mo Soilh USH El li Ezzor lgir lgri rizg@gmilom USH El li Ezzor lgir lgri moilh@gmilom r In hi r h glol in of h olion o h rion iffion m SK inro Shg Kzki rmoo mol [] in olion nmi ih iniil oniion n homogno Nmnn on onir h SK mol iho mi iffion gi ffiin oniion on h rion rmr o nr h glol in of h olion o h rolm in h Sool In rm:rion iffion m glol in olion nmi hg kki rmoo mol 499

2 h o Si Mol onirh Shg Kki rmoo mol [] for o i or o hmil n in rion liing in on omin of N R N 3 of il nii im n n i h ni or norml o f > g > > n n hr f g i i i i i i i r l oii onn rrning h iffion n rion rmr of h mol hir iologil n ologil ignifinion i ll knon for ml [][8][9][] N i h lin il oror i i If i i h hihoril Volrr mol If i i i i i h ok-volrr m i i ri h iffion of h olion in ri h lf iffion of h olion ri h mi iffion of h olion in of h r h in of h lol n glol olion o h m SK iho mi iffion i f g n n ] [ ] [ ] [ ol Ein of h Solion in In h folloing ill h mho of [4] ill h horm 63 onir h m ih rion f g h iffion n rin 5

3 h om g f 3 Dfin h oror n h m 3 k h form D g D f 4 In hi ion onir h ilir qilinr m ; U DU U U U U hr U ih m n g f DU U n fin for / > on : n n for : > / horm So h r oii iniil oniion n N hr i > h h h m h lol olion ] ] ] ] Proofo ro h in n niqn of h lol olion ill imlmn h fi oniion Q-Q5 of [] follo : Q hoo h hr follo 5

4 R R h R R n P hrfor R R Q oring o h horm8 of [3] fin h inrolion o h horm of [] hr i onn > n o g Q3or U U fin riol if R h ro for h i n lmn of V hrv i h on of fin R V If S i mri n nh no S h of fnion f onino onu h h for h V n f niforml -Holr onino on V or S f hi rqir ' ' ' ' f f S ' ' V hr R hi imion om U hr 5

5 oringl for V > > N R R Q4 h horm 66 of [] hr R R from hih g for >> lo h > oring o h horm 66 of [] h oror ill rongl niforml roli Q5 h Prooiion56 of [3] U DU f g N hr i hiing n R R 5 o l h Prooiion 56 of [3] m rif o oniion: I no h i of h form f g G n hr G i n on G R R n of h form n n o g h n G R f g II i no ning of n if G R 53

6 n R R R 3 Glol Ein of h Solion h o ro h in of h glol olion o h m iho mi iffion mho inir from h on in [] for lol olion of qilinr roli m hi hor nr h in of h olion for ll n llo o mo from h lol olion o h glol on h i oni of roing h in of onn ning on n ifing : > > no horm or oii iniil oniion hr i glol olion of h m nr h oniion > > of h rion rmr Proofo ro h glol n of h olion ill ii h roof ino hr S : Poiii hn of in h Sool ir S S : S 3 : So h on ill h fnion m m Mlil h fir qion of h m 4 n ingr or o g 3 In hi rlion n of h Sool ming of h in if on h 54

7 nr h oniion on > / N > N > 3/ N 3 6 h oringl 7 8 rom 7 8 g 9 h m hrfor Uing Gronll lmm in 9 h rlion imli nr h oniion hih ro h oiii of h olion h m mho li o on qion of h m mlili hn ingr or n ing h oniion o g Son S Mliling h o qion of h m ril hn ingr or g 3 55

8 g n n 4 Uing h onr oniion n n on hn45 om 5 omin 54 n 3 hn 6 or n > h 7 Uing h oiii in n n h rlion 56

9 8 Uing 8 in 7 9 m o h oniion n ing hm in 9 o oin 57

10 Soling h iffrnil inqion g P P P P hir S ro h in of oii onn oning h olion in R o hi n h o rio ll om on of oror Rri h m follo k k k k 3 hr > k i ffiinl lrg or Dfin Q Q Q n k k rom h oiii h > > Q Q h onn k i hon ffiinl lrg n l h inqli 638 in g 76 of [6] in o g h ingrl inqion 4 for n > hr h oror 58

11 k Q k Q r finfor oring o h rfrn [6] n [7] in horm 6 h D n oring o [5] g 4 h D h o inqion 4 om 5 hr k k 6 n > from oin 7 Uing h gmm fnion h for 8 Uing 7 n 8 in 5 n > 9 No ro h m rl for h 3 I i fin for oring o [5] g 4 h D rom h o 3 gi 3 59

12 h k k n ing 9 h rlion 3 om for oringl ing 7 n 8 h rlion 5 om for n finll l h horm in g 54 of rfrn [] o g h glol olion of h rolm Rfrn [] mmn H Qi linr Proli Sm nr Nonlinr onr oniion rn mr Mh So [] mmn H Glol in for mi linr roli m in Nonlinr nli: ollion of Pr in Honor of Erih H Roh ri R Knnn n H R inrgr mi Pr N ork [3] mmn H Ein n rglri for mi linr roli olion qion nn Sol Norm S Pi Si [4] mmn H Qilinr olion qion n roli m rn r Mh So [5] mmn H On r roli fnmnl olion J Mh So Jn [6] rimn Pril Diffrnil Eqion Hol Rinhr n inon N ork 969 [7] Hnr D Gomri hor of mi linr roli qion Sringr-Vrlg r No in Mhmi [8] Kohi S ong KE Prh RD Glol in for rongl ol rion iffion m 4 [9] D Ngn Glol in for l of ringlr roli m on omin of rirr imnion MS

13 [] D Ngn V Ngn Shig-Kki- rmoo mol on highr imnionl omin Elroni Jornl of Diffrnil Eqion 7 3 [] Shig N Kki K rmoo E Sil grgion of inring i J hor iol

14 5

Mathcad Lecture #4 In-class Worksheet Vectors and Matrices 1 (Basics)

Mathcad Lecture #4 In-class Worksheet Vectors and Matrices 1 (Basics) Mh Lr # In-l Workh Vor n Mri (Bi) h n o hi lr, o hol l o: r mri n or in Mh i mri prorm i mri mh oprion ol m o linr qion ing mri mh. Cring Mri Thr r rl o r mri. Th "Inr Mri" Wino (M) B K Poin Rr o