https://www.chenk.top/en/tags/lotka-volterra/Recent content in Lotka-Volterra on Chen Kai BlogHugoenSat, 24 Feb 2024 09:00:00 +0000https://www.chenk.top/en/ode/15-population-dynamics/Sat, 24 Feb 2024 09:00:00 +0000https://www.chenk.top/en/ode/15-population-dynamics/<p><strong>Why do lynx and snowshoe hare populations cycle with eerie regularity over a 10-year period?</strong> Why does introducing a single new species sometimes collapse an entire ecosystem? Why do similar competitors sometimes coexist and sometimes drive each other extinct? The answers are not in the species; they are in the <em>equations</em> relating the species. This chapter walks through the canonical models of mathematical ecology: from the single-population logistic and Allee models to multi-species competition, predator-prey oscillations, age structure, and spatial spread.</p>https://www.chenk.top/en/ode/08-nonlinear-stability/Sat, 28 Oct 2023 09:00:00 +0000https://www.chenk.top/en/ode/08-nonlinear-stability/<p><strong>The real world is nonlinear.</strong> Predator-prey cycles, heartbeat rhythms, neuron firing — none of these can be captured by linear equations. When superposition fails, the world acquires <em>new</em> behaviors: limit cycles, multiple equilibria, bistability, hysteresis. This chapter gives you the geometric and analytic tools to read those behaviors directly off a 2D phase portrait.</p> <p><figure class="article-figure"> <img src="https://blog-pic-ck.oss-cn-beijing.aliyuncs.com/posts/en/ode/08-nonlinear-stability/illustration_1.png" alt="Ordinary Differential Equations (8): Nonlinear Systems and Phase Portraits — Chapter overview" loading="lazy" decoding="async" class="content-image"> </figure> </p> <hr> <h2 id="what-you-will-learn" class="heading-anchor">What You Will Learn<a href="#what-you-will-learn" class="heading-link" aria-label="Permalink to this section" title="Copy link to this section">#</a> </h2><ul> <li>Why nonlinear systems are <em>fundamentally</em> different from linear ones</li> <li>Lyapunov stability visualized: level sets, bowls, and basins</li> <li>Linearization vs. the full nonlinear picture (Hartman-Grobman in action)</li> <li>Lotka-Volterra predator-prey: closed orbits and conserved quantities</li> <li>Competition models: four canonical outcomes</li> <li>Van der Pol oscillator and the geometry of limit cycles</li> <li>Gradient and Hamiltonian systems</li> <li>Poincaré-Bendixson: why 2D systems cannot be chaotic</li> </ul> <h2 id="prerequisites" class="heading-anchor">Prerequisites<a href="#prerequisites" class="heading-link" aria-label="Permalink to this section" title="Copy link to this section">#</a> </h2><ul> <li><a href="https://www.chenk.top/en/ode/06-power-series/">Chapter 6</a> : linear systems, phase portrait classification</li> <li><a href="https://www.chenk.top/en/ode/07-systems-and-phase-plane/">Chapter 7</a> : stability, linearization, Lyapunov functions</li> </ul> <hr> <h2 id="from-linear-to-nonlinear" class="heading-anchor">From Linear to Nonlinear<a href="#from-linear-to-nonlinear" class="heading-link" aria-label="Permalink to this section" title="Copy link to this section">#</a> </h2><p>Linear systems obey <strong>superposition</strong>: if <span class="math-inline">$\mathbf{x}_1$</span> and <span class="math-inline">$\mathbf{x}_2$</span> are solutions, so is <span class="math-inline">$c_1\mathbf{x}_1 &#43; c_2\mathbf{x}_2$</span> . This is the engine that powers the entire toolkit of Chapters 1-6 — exponential ansatz, eigenvectors, fundamental matrices.</p>