On a 2D Model of Avascular Tumor with Weak Allee Effect

<figure class="figure-group responsive"><div class="sub-figure" id="a"><object data="https://static.hindawi.com/articles/jam/volume-2019/9581072/figures/9581072.fig.0013a.svgz" name="9581072.fig.0013a" type="image/svg+xml"></object><div class="fig-label">(a) Proliferating cells</div></div><div class="sub-figure" id="b"><object data="https://static.hindawi.com/articles/jam/volume-2019/9581072/figures/9581072.fig.0013b.svgz" name="9581072.fig.0013b" type="image/svg+xml"></object><div class="fig-label">(b) Quiescent cells</div></div><div class="sub-figure" id="c"><object data="https://static.hindawi.com/articles/jam/volume-2019/9581072/figures/9581072.fig.0013c.svgz" name="9581072.fig.0013c" type="image/svg+xml"></object><div class="fig-label">(c) Necrotic cells</div></div><div class="sub-figure" id="d"><object data="https://static.hindawi.com/articles/jam/volume-2019/9581072/figures/9581072.fig.0013d.svgz" name="9581072.fig.0013d" type="image/svg+xml"></object><div class="fig-label">(d) Proliferating cells</div></div><div class="sub-figure" id="e"><object data="https://static.hindawi.com/articles/jam/volume-2019/9581072/figures/9581072.fig.0013e.svgz" name="9581072.fig.0013e" type="image/svg+xml"></object><div class="fig-label">(e) Quiescent cells</div></div><div class="sub-figure" id="f"><object data="https://static.hindawi.com/articles/jam/volume-2019/9581072/figures/9581072.fig.0013f.svgz" name="9581072.fig.0013f" type="image/svg+xml"></object><div class="fig-label">(f) Necrotic cells</div></div></figure>

<p>Heat map of cell densities at different proliferating rates: the top row corresponds to the standard rate at <span class="nowrap"><svg height="12.7178pt" id="M154" style="vertical-align:-3.42947pt" version="1.1" viewbox="-0.0498162 -9.28833 38.1062 12.7178" width="38.1062pt" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M558 587C558 666 497 712 432 712C379 712 330 691 284 650C212 586 178 508 148 348L71 -65C49 -185 35 -229 23 -235L27 -261C49 -259 101 -251 124 -224C131 -216 138 -178 159 -24C171 66 197 200 227 356C264 550 295 668 393 668C443 668 479 632 479 575C479 516 446 458 383 418C361 404 344 398 318 397L296 350C395 338 460 281 460 192C460 110 411 40 300 40C258 40 215 55 192 69L181 51C191 18 222 -16 266 -16C308 -16 351 1 397 26C471 67 545 142 545 221C545 315 486 365 401 395C469 437 558 498 558 587Z"></path></g><g transform="matrix(.013,0,0,-0.013,11.186,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,22.449,0)"><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,-0.013,28.689,0)"><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,-0.013,31.653,0)"><path d="M456 178V225H360V632H320C217 496 115 347 20 206V178H280V106C280 40 276 34 189 27V0H445V27C364 34 360 39 360 106V178H456ZM280 225H82C149 335 214 431 278 520H280V225Z"></path></g></svg>;</span> the bottom row corresponds to a higher proliferating rate at <span class="nowrap"><svg height="12.7178pt" id="M155" style="vertical-align:-3.42947pt" version="1.1" viewbox="-0.0498162 -9.28833 38.1062 12.7178" width="38.1062pt" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M558 587C558 666 497 712 432 712C379 712 330 691 284 650C212 586 178 508 148 348L71 -65C49 -185 35 -229 23 -235L27 -261C49 -259 101 -251 124 -224C131 -216 138 -178 159 -24C171 66 197 200 227 356C264 550 295 668 393 668C443 668 479 632 479 575C479 516 446 458 383 418C361 404 344 398 318 397L296 350C395 338 460 281 460 192C460 110 411 40 300 40C258 40 215 55 192 69L181 51C191 18 222 -16 266 -16C308 -16 351 1 397 26C471 67 545 142 545 221C545 315 486 365 401 395C469 437 558 498 558 587Z"></path></g><g transform="matrix(.013,0,0,-0.013,11.186,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,22.449,0)"><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,-0.013,28.689,0)"><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,-0.013,31.653,0)"><path d="M137 343C167 482 260 545 321 574C357 591 397 603 429 609L423 641C382 634 335 622 295 608C189 570 37 457 37 238C37 84 125 -12 242 -12C362 -12 447 89 447 209C447 311 374 393 267 393C247 393 226 386 204 376L137 343ZM227 337C318 337 361 256 361 173C361 105 336 22 258 22C176 22 126 120 126 240C126 266 127 291 132 310C155 323 189 337 227 337Z"></path></g></svg>.</span></p>

Journal of Applied Mathematics

On a 2D Model of Avascular Tumor with Weak Allee Effect

Figure 13