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Community matrix

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575: 190: 358: 475:
Berlow, E. L.; Neutel, A.-M.; Cohen, J. E.; De Ruiter, P. C.; Ebenman, B.; Emmerson, M.; Fox, J. W.; Jansen, V. A. A.; Jones, J. I.; Kokkoris, G. D.; Logofet, D. O.; McKane, A. J.; Montoya, J. M; Petchey, O. (2004).
60: 249: 444: 410: 185:{\displaystyle {\begin{array}{rcl}{\dfrac {dx}{dt}}&=&x(\alpha -\beta y)\\{\dfrac {dy}{dt}}&=&-y(\gamma -\delta x),\end{array}}} 635: 616: 32: 353:{\displaystyle {\begin{bmatrix}{\frac {du}{dt}}\\{\frac {dv}{dt}}\end{bmatrix}}=\mathbf {A} {\begin{bmatrix}u\\v\end{bmatrix}},} 556: 446:
have positive real part then the equilibrium is unstable, but if all eigenvalues have negative real part then it is stable.
655: 528: 609: 51: 640: 228: 645: 602: 427: 393: 421: 518: 455: 232: 20: 590: 8: 582: 547:, Interdisciplinary Applied Mathematics, vol. 17 (3rd ed.), Berlin, New York: 499: 44: 65: 650: 552: 524: 494: 477: 36: 489: 548: 388: 586: 629: 28: 503: 40: 574: 478:"Interaction Strengths in Food Webs: Issues and Opportunities" 235:
to a linearization of the system about an equilibrium point (
474: 326: 258: 430: 396: 252: 123: 69: 63: 438: 404: 352: 184: 627: 610: 523:. Cambridge University Press. p. 144. 617: 603: 420:*) is called the community matrix. By the 493: 545:Mathematical Biology I. An Introduction 628: 542: 43:of the community matrix determine the 636:Mathematical and theoretical biology 569: 412:evaluated at the equilibrium point ( 516: 33:generalized Lotkaā€“Volterra equation 13: 52:Lotkaā€“Volterra predatorā€“prey model 14: 667: 573: 520:Elements of Mathematical Ecology 495:10.1111/j.0021-8790.2004.00833.x 432: 424:, if one or both eigenvalues of 398: 387:*. In mathematical biology, the 317: 211:) the number of predators, and 510: 468: 203:) denotes the number of prey, 172: 157: 115: 100: 1: 461: 589:. You can help Knowledge by 439:{\displaystyle \mathbf {A} } 405:{\displaystyle \mathbf {A} } 7: 449: 10: 672: 568: 47:of the equilibrium point. 656:Applied mathematics stubs 543:Murray, James D. (2002), 482:Journal of Animal Ecology 231:the non-linear system is 233:topologically equivalent 422:stable manifold theorem 243:*), which has the form 229:Hartmanā€“Grobman theorem 585:-related article is a 440: 406: 354: 227:are constants. By the 186: 456:Paradox of enrichment 441: 407: 355: 187: 428: 394: 250: 61: 21:mathematical biology 583:applied mathematics 641:Population ecology 517:Kot, Mark (2001). 436: 402: 350: 341: 307: 182: 180: 142: 88: 646:Dynamical systems 598: 597: 558:978-0-387-95223-9 303: 279: 141: 87: 50:For example, the 37:equilibrium point 663: 619: 612: 605: 577: 570: 561: 535: 534: 514: 508: 507: 497: 472: 445: 443: 442: 437: 435: 411: 409: 408: 403: 401: 359: 357: 356: 351: 346: 345: 320: 312: 311: 304: 302: 294: 286: 280: 278: 270: 262: 191: 189: 188: 183: 181: 143: 140: 132: 124: 89: 86: 78: 70: 25:community matrix 16:Community Matrix 671: 670: 666: 665: 664: 662: 661: 660: 626: 625: 624: 623: 566: 559: 549:Springer-Verlag 539: 538: 531: 515: 511: 473: 469: 464: 452: 431: 429: 426: 425: 397: 395: 392: 391: 389:Jacobian matrix 340: 339: 333: 332: 322: 321: 316: 306: 305: 295: 287: 285: 282: 281: 271: 263: 261: 254: 253: 251: 248: 247: 179: 178: 149: 144: 133: 125: 122: 119: 118: 95: 90: 79: 71: 68: 64: 62: 59: 58: 17: 12: 11: 5: 669: 659: 658: 653: 648: 643: 638: 622: 621: 614: 607: 599: 596: 595: 578: 564: 563: 557: 537: 536: 529: 509: 488:(5): 585ā€“598. 466: 465: 463: 460: 459: 458: 451: 448: 434: 400: 361: 360: 349: 344: 338: 335: 334: 331: 328: 327: 325: 319: 315: 310: 301: 298: 293: 290: 284: 283: 277: 274: 269: 266: 260: 259: 257: 193: 192: 177: 174: 171: 168: 165: 162: 159: 156: 153: 150: 148: 145: 139: 136: 131: 128: 121: 120: 117: 114: 111: 108: 105: 102: 99: 96: 94: 91: 85: 82: 77: 74: 67: 66: 15: 9: 6: 4: 3: 2: 668: 657: 654: 652: 649: 647: 644: 642: 639: 637: 634: 633: 631: 620: 615: 613: 608: 606: 601: 600: 594: 592: 588: 584: 579: 576: 572: 571: 567: 560: 554: 550: 546: 541: 540: 532: 530:0-521-00150-1 526: 522: 521: 513: 505: 501: 496: 491: 487: 483: 479: 471: 467: 457: 454: 453: 447: 423: 419: 415: 390: 386: 382: 378: 374: 370: 366: 347: 342: 336: 329: 323: 313: 308: 299: 296: 291: 288: 275: 272: 267: 264: 255: 246: 245: 244: 242: 238: 234: 230: 226: 222: 218: 214: 210: 206: 202: 198: 175: 169: 166: 163: 160: 154: 151: 146: 137: 134: 129: 126: 112: 109: 106: 103: 97: 92: 83: 80: 75: 72: 57: 56: 55: 53: 48: 46: 42: 38: 34: 30: 29:linearization 26: 22: 591:expanding it 580: 565: 544: 519: 512: 485: 481: 470: 417: 413: 384: 380: 376: 372: 368: 364: 362: 240: 236: 224: 220: 216: 212: 208: 204: 200: 196: 194: 49: 24: 18: 41:eigenvalues 630:Categories 462:References 167:δ 164:− 161:γ 152:− 110:β 107:− 104:α 45:stability 651:Matrices 450:See also 504:3505669 27:is the 555:  527:  502:  375:* and 363:where 195:where 39:. The 35:at an 23:, the 581:This 500:JSTOR 31:of a 587:stub 553:ISBN 525:ISBN 223:and 490:doi 416:*, 239:*, 54:is 19:In 632:: 551:, 498:. 486:73 484:. 480:. 383:āˆ’ 379:= 371:āˆ’ 367:= 219:, 215:, 618:e 611:t 604:v 593:. 562:. 533:. 506:. 492:: 433:A 418:y 414:x 399:A 385:y 381:y 377:v 373:x 369:x 365:u 348:, 343:] 337:v 330:u 324:[ 318:A 314:= 309:] 300:t 297:d 292:v 289:d 276:t 273:d 268:u 265:d 256:[ 241:y 237:x 225:Ī“ 221:Ī³ 217:Ī² 213:Ī± 209:t 207:( 205:y 201:t 199:( 197:x 176:, 173:) 170:x 158:( 155:y 147:= 138:t 135:d 130:y 127:d 116:) 113:y 101:( 98:x 93:= 84:t 81:d 76:x 73:d

Index

mathematical biology
linearization
generalized Lotkaā€“Volterra equation
equilibrium point
eigenvalues
stability
Lotkaā€“Volterra predatorā€“prey model
Hartmanā€“Grobman theorem
topologically equivalent
Jacobian matrix
stable manifold theorem
Paradox of enrichment
"Interaction Strengths in Food Webs: Issues and Opportunities"
doi
10.1111/j.0021-8790.2004.00833.x
JSTOR
3505669
Elements of Mathematical Ecology
ISBN
0-521-00150-1
Springer-Verlag
ISBN
978-0-387-95223-9
Stub icon
applied mathematics
stub
expanding it
v
t
e

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