Codimension-2 Bifurcation Analysis and Control of a Discrete Mosquito Model with a Proportional Release Rate of Sterile Mosquitoes

<div>1 : 3 strong resonance bifurcation diagram at (0.61636576, 0.605198): (a) in <svg height="11.5564pt" id="M287" style="vertical-align:-2.26807pt" version="1.1" viewbox="-0.0498162 -9.28833 38.4933 11.5564" width="38.4933pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,4.498,0)"><path d="M483 97L471 123C436 91 401 65 392 65C388 65 384 74 390 106C414 239 444 378 457 429L455 433C444 433 429 436 416 439C392 444 368 448 344 448C281 448 204 415 152 376C71 315 23 205 23 103C23 21 57 -12 85 -12C114 -12 149 6 185 34C231 70 285 119 329 183H331L309 81C292 0 308 -12 326 -12C350 -12 421 24 483 97ZM374 387C370 363 356 291 345 261C315 193 181 50 139 50C124 50 110 71 110 118C110 224 153 331 218 379C238 394 271 402 301 402C329 402 359 394 374 387Z"></path></g><g transform="matrix(.013,0,0,-0.013,13.984,0)"><path d="M535 230V280H52V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,24.52,0)"><path d="M689 332C689 394 670 448 646 448C620 448 597 421 597 396C597 386 600 381 608 372C619 359 620 334 620 315C620 150 538 45 454 45C414 45 386 67 386 122C386 138 388 158 394 180L457 426L452 432L377 416L315 156C302 100 259 45 216 45C176 45 148 67 148 122C148 133 152 158 156 180C162 212 173 259 194 332C201 357 206 384 206 405C206 430 198 448 174 448C125 448 66 406 23 342L43 319C84 368 110 383 121 383C126 383 128 382 128 377C128 370 127 359 122 343C99 268 84 204 77 156C74 137 70 111 70 104C70 25 125 -12 180 -12C228 -12 276 12 319 50C338 8 378 -12 418 -12C549 -12 689 166 689 332Z"></path></g><g transform="matrix(.013,0,0,-0.013,33.777,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g></svg> plane with <svg height="9.49473pt" id="M288" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -9.28833 80.4885 9.49473" width="80.4885pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M452 333C452 398 425 448 375 448C329 448 235 404 140 292H138L229 683C233 701 234 712 225 712C208 712 149 686 66 677L64 652H97C135 652 140 651 129 598L38 167C23 96 29 57 44 33C60 7 98 -12 132 -12C169 -12 232 3 290 41C383 102 452 223 452 333ZM365 316C365 199 308 87 239 49C221 39 200 35 183 35C137 35 99 70 112 163C116 191 123 215 129 233C190 316 290 392 330 392C348 392 365 372 365 316Z"></path></g><g transform="matrix(.013,0,0,-0.013,9.733,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,20.996,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"></path></g><g transform="matrix(.013,0,0,-0.013,27.236,0)"><path d="M153 550H386L412 615L406 623H120L82 318C104 327 142 338 184 338C294 338 347 275 347 187C347 112 305 39 221 39C160 39 119 71 97 89C88 97 80 96 71 90C59 80 50 67 49 57C48 45 52 36 66 23C80 9 123 -12 169 -12C221 -11 288 15 342 59C403 109 431 165 431 225C431 308 366 395 238 395C212 395 165 379 127 364L153 550Z"></path></g><g transform="matrix(.013,0,0,-0.013,33.476,0)"><path d="M113 -12C146 -12 170 11 170 46C170 78 146 103 114 103S58 78 58 46C58 11 82 -12 113 -12Z"></path></g><g transform="matrix(.013,0,0,-0.013,36.44,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"></path></g><g transform="matrix(.013,0,0,-0.013,42.68,0)"><path d="M153 550H386L412 615L406 623H120L82 318C104 327 142 338 184 338C294 338 347 275 347 187C347 112 305 39 221 39C160 39 119 71 97 89C88 97 80 96 71 90C59 80 50 67 49 57C48 45 52 36 66 23C80 9 123 -12 169 -12C221 -11 288 15 342 59C403 109 431 165 431 225C431 308 366 395 238 395C212 395 165 379 127 364L153 550Z"></path></g><g transform="matrix(.013,0,0,-0.013,48.92,0)"><path d="M447 623H65C61 580 56 530 47 475H76C100 541 106 550 172 550H388C308 376 196 170 91 -1L98 -12L172 -2C268 204 360 408 455 611L447 623Z"></path></g><g transform="matrix(.013,0,0,-0.013,55.163,0)"><path d="M153 550H386L412 615L406 623H120L82 318C104 327 142 338 184 338C294 338 347 275 347 187C347 112 305 39 221 39C160 39 119 71 97 89C88 97 80 96 71 90C59 80 50 67 49 57C48 45 52 36 66 23C80 9 123 -12 169 -12C221 -11 288 15 342 59C403 109 431 165 431 225C431 308 366 395 238 395C212 395 165 379 127 364L153 550Z"></path></g><g transform="matrix(.013,0,0,-0.013,61.403,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z"></path></g><g transform="matrix(.013,0,0,-0.013,67.643,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"></path></g><g transform="matrix(.013,0,0,-0.013,73.883,0)"><path d="M447 623H65C61 580 56 530 47 475H76C100 541 106 550 172 550H388C308 376 196 170 91 -1L98 -12L172 -2C268 204 360 408 455 611L447 623Z"></path></g></svg> and (b) in <svg height="11.5564pt" id="M289" style="vertical-align:-2.26807pt" version="1.1" viewbox="-0.0498162 -9.28833 58.0838 11.5564" width="58.0838pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,4.498,0)"><path d="M452 333C452 398 425 448 375 448C329 448 235 404 140 292H138L229 683C233 701 234 712 225 712C208 712 149 686 66 677L64 652H97C135 652 140 651 129 598L38 167C23 96 29 57 44 33C60 7 98 -12 132 -12C169 -12 232 3 290 41C383 102 452 223 452 333ZM365 316C365 199 308 87 239 49C221 39 200 35 183 35C137 35 99 70 112 163C116 191 123 215 129 233C190 316 290 392 330 392C348 392 365 372 365 316Z"></path></g><g transform="matrix(.013,0,0,-0.013,13.505,0)"><path d="M535 230V280H52V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,24.041,0)"><path d="M483 97L471 123C436 91 401 65 392 65C388 65 384 74 390 106C414 239 444 378 457 429L455 433C444 433 429 436 416 439C392 444 368 448 344 448C281 448 204 415 152 376C71 315 23 205 23 103C23 21 57 -12 85 -12C114 -12 149 6 185 34C231 70 285 119 329 183H331L309 81C292 0 308 -12 326 -12C350 -12 421 24 483 97ZM374 387C370 363 356 291 345 261C315 193 181 50 139 50C124 50 110 71 110 118C110 224 153 331 218 379C238 394 271 402 301 402C329 402 359 394 374 387Z"></path></g><g transform="matrix(.013,0,0,-0.013,33.527,0)"><path d="M535 230V280H52V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,44.063,0)"><path d="M689 332C689 394 670 448 646 448C620 448 597 421 597 396C597 386 600 381 608 372C619 359 620 334 620 315C620 150 538 45 454 45C414 45 386 67 386 122C386 138 388 158 394 180L457 426L452 432L377 416L315 156C302 100 259 45 216 45C176 45 148 67 148 122C148 133 152 158 156 180C162 212 173 259 194 332C201 357 206 384 206 405C206 430 198 448 174 448C125 448 66 406 23 342L43 319C84 368 110 383 121 383C126 383 128 382 128 377C128 370 127 359 122 343C99 268 84 204 77 156C74 137 70 111 70 104C70 25 125 -12 180 -12C228 -12 276 12 319 50C338 8 378 -12 418 -12C549 -12 689 166 689 332Z"></path></g><g transform="matrix(.013,0,0,-0.013,53.32,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g></svg> space.</div>

Complexity

fig3

Figure 3

Figure 3: Codimension-2 Bifurcation Analysis and Control of a Discrete Mosquito Model with a Proportional Release Rate of Sterile Mosquitoes 