Mathematical biology

Mathematical biology uses mathematical models to connect biological mechanisms with quantitative predictions. Differential equations can track changing quantities, while population dynamics, enzyme kinetics and gene regulatory networks ask how growth, reactions and feedback shape living systems. A useful mathematical model states its assumptions and units: a simple chemostat, for example, reveals how nutrient supply and removal jointly determine whether a culture persists.

The system $\dot{u}=u(1-u)-auv/(u+d)$, $\dot{v}=bv(1-v/u)$ combines logistic prey growth, saturating predation and a predator carrying level proportional to prey density. Its unique positive equilibrium is $u=v=s$, where $(1-s)(s+d)=as$. Its Jacobian matrix has determinant $bs\sqrt{(1-a-d{)}^2+4d}/(s+d)>0$ and trace $s(1-d-2s)/(s+d)-b$. The positive trapping construction for the Leslie-Gower model gives a compact positive region and a limit cycle whenever this equilibrium is a repeller.

A rectangular upper bound $u,v\le M$, $M>1$, is inward pointing for the Leslie-Gower predator-prey model. To avoid the singular edge $u=0$, use $P=u{e}^{-v}$ and the inequality $\dot{P}/P\ge 1-u[1+(a/d+b{)}^2/(4b)]$, obtained by completing a square. Choose $\epsilon e^M$ sufficiently small and $0<\eta <\epsilon$. Then $u\le M$, $\eta \le v\le M$, $P\ge \epsilon$ is a compact forward-invariant set in the positive quadrant. Removing an outward-flow neighborhood of a repelling equilibrium leaves a trapping region without equilibria, so the Poincaré-Bendixson theorem yields a periodic omega-limit orbit.

In a biological competition model, increasing one population reduces the growth of another. A two-species example is $\dot{u}=ru(1-u-v)$, $\dot{v}=v(1-\beta u)$, with both cross-effects negative. This asymmetric example has self-limitation only in $u$; disappearance of $u$ therefore permits unbounded growth of $v$. For $\beta >1$ its positive coexistence equilibrium is a saddle whose stable manifold separates exclusion outcomes.

For the population map $f(N)=\alpha N^2/(\beta N^2+S)$, extinction is a superstable fixed point. Below $S_c$, two positive fixed points $N_{\pm }=(\alpha \pm \sqrt{{\alpha }^2-4\beta S})/(2\beta )$ exist. Their multipliers are $1\mp \sqrt{{\alpha }^2-4\beta S}/\alpha$, making the lower one a threshold and the upper one attracting. A saddle-node bifurcation removes them at $S_c$; above it $f(N)<N$ for every positive population, so every nonnegative orbit tends to extinction.

The mathematical description of elastic bodies whose stress-free material state changes through growth or remodelling. A multiplicative decomposition of the deformation gradient separates the growth tensor $\mathbf{G}$ from the elastic accommodation $\mathbf{A}$. The relaxed grown configuration need not be globally compatible, so even an unloaded body can develop internal stress.

For $A,B,\rho >0$, these Lotka-Volterra equations describe two populations limited by their own density and depressed by the other population. The nonnegative quadrant is invariant. Positive equilibria are intersections of the non-axis nullclines $u+Av=a$ and $Bu+v=b$; their existence and stability distinguish stable coexistence, competitive exclusion, and bistability. Positive solutions have no periodic orbits: multiplying the vector field by the Dulac function $1/(uv)$ makes its divergence $-1/v-\rho /u<0$, so the Bendixson-Dulac theorem applies. Boundedness, boundary equilibria, and their stable manifolds must still be checked before concluding convergence to one particular equilibrium.

For $\dot{n}=n(1-n-m/2)$, $\dot{m}=qm(1-m-n/2)$, choose $k=2/3$ and weights $1,1/q$. Then $\dot{V}=-(n-k{)}^2-(m-k{)}^2-(n-k)(m-k)$, strictly negative away from coexistence. The function diverges at the positive-quadrant boundary and at infinity, keeping trajectories in compact interior level sets. This proves global convergence for positive initial states, while boundary-axis data retain their single-species dynamics.

The FitzHugh-Nagumo model represents an excitable cell by a fast activation variable and a slower recovery variable. A common nondimensional form is $\dot{V}=V-V^3/3-W+I$, $\dot{W}=\epsilon (V+a-bW)$. Its cubic activation response and recovery feedback can produce an excitable resting state or sustained oscillations through bifurcations. Particular reductions and travelling-wave equations can have additional state variables; their signs and parameter conventions must be stated separately.

For the scalar reaction-diffusion equation $v_t=D{v}_{xx}-(v-r_1)(v-r_2)(v-r_3)$ with $D>0$ and $r_1<r_2<r_3$, an exact decreasing travelling wave isIndeed $V^{\prime }=-(V-r_1)(r_3-V)/\sqrt{2D}$, and substitution into $D{V}^{\prime \prime }+c{V}^{\prime }-(V-r_1)(V-r_2)(V-r_3)=0$ leaves the factor $(r_1+r_3)/2-r_2-c/\sqrt{2D}$, proving the speed formula. A zero speed gives the Maxwell balance for a scalar reaction-diffusion front. In a FitzHugh-Nagumo model this calculation requires treating the recovery variable as frozen externally: retaining $w_t=\beta v-\gamma w$ with exactly constant $w$ forces $v$ to be constant.

A recovery variable represents feedback that reduces activation after an excitation in a FitzHugh-Nagumo model. Its equation must be specified separately from the activation equation. Holding it externally constant gives a scalar reduction; requiring it to remain constant while keeping its original evolution equation generally imposes an additional constraint and need not permit the same solutions.

For $\dot{v}=v(\alpha -v)(v-1)-w+I$, $\dot{w}=\beta v-\gamma w$, set $k=\beta /\gamma$. Intersections of the nullclines solve $I=h(v)=v^3-(1+\alpha )v^2+(\alpha +k)v$. Three simple equilibria are possible precisely when $k<M=(1-\alpha +{\alpha }^2)/3$ and $h(u_{+})<I<h(u_{-})$, where $u_{\pm }=(1+\alpha \pm \sqrt{1-\alpha +{\alpha }^2-3k})/3$. To prove this, observe that $h^{\prime }=k-M+3(v-(1+\alpha )/3{)}^2$ has two real zeros exactly when $k<M$; monotonicity on the three resulting intervals counts the intersections. At either endpoint there are a double and a simple root, and at $k=M$, $I=h((1+\alpha )/3)$ there is one triple root.

An epidemic is a period of growing infection incidence in a population. Compartmental models describe transmission, recovery and depletion or renewal of susceptible hosts. An epidemic invasion threshold concerns growth after a small introduction; a final size relation for an epidemic concerns the total susceptible depletion of a closed outbreak. These are distinct from an endemic equilibrium with continuing transmission.

A metapopulation describes occupied habitat patches with local colonization and extinction. The patch occupancy variable differs from population abundance within a patch.

This model couples deterministic underdominant allele-frequency dynamics in equal-sized habitats by global migration. Starting with $m$ pure habitats of one allele and $N-m$ of the other preserves two within-class frequencies $x,y$, reducing the system to $\dot{x}=f(x)-\sigma (1-q)(x-y)$, $\dot{y}=f(y)+\sigma q(x-y)$ with $q=m/N$. Migration is conservative in the mean, but selection need not conserve global allele frequency.

With equally many initially pure habitats of each allele, symmetry preserves $y=1-x$. The heterogeneous equilibrium exists for $\sigma <s/2$. Its full-system eigenvalues are $-s+3\sigma$ in the mean-frequency direction and $-s+2\sigma$ in the antisymmetric direction, so it attracts general nearby perturbations only for $\sigma <s/3$. Exact symmetry can preserve both alleles beyond this robust range; at stronger migration it approaches the unstable uniform midpoint.

In the two-class globally coupled underdominant metapopulation, the rectangle $3/4\le x\le 1$, $0\le y\le 1/4$ is forward invariant under the displayed bound. On its inner edges, selection points inward with magnitude $3s/32$ while outward migration is at most $(3/4)\sigma (1-q)$ or $(3/4)\sigma q$; the outer edges also point inward. Thus prescribed initial state $(1,0)$ cannot approach global allele fixation when $0<q<1$. Maximizing $f(u)/u=s(1-u)(2u-1)$ gives $s/8$ at $u=3/4$, making the bound optimal within this particular symmetric worst-case rectangle argument, not a universal sharp migration threshold.

Colonization of empty habitat at rate $c$ and extinction of occupied habitat at rate $m$ give the displayed patch-occupancy equation. The positive equilibrium is $1-m/c$ when $c>m$. Competition can be incorporated through replacement fluxes between patches occupied by different species.

A resident trait $x$ occupies fraction $1-m/c(x)$ when its colonization rate exceeds fire extinction. With net competitive replacement given by ability difference $B(y)-B(x)$, a rare mutant has the displayed per-capita growth. The selection gradient is $g(x)=m{c}^{\prime }(x)/c(x)+[1-m/c(x)]B^{\prime }(x)$. A singular trait with negative mutant-fitness curvature is a strict local evolutionarily stable strategy. Global strict uninvadability requires that $mc(y)+[c(x)-m]B(y)$ be smaller than its resident value for every other trait. Monotonicity alone does not specify the shape or existence of an optimum.

An Allee effect makes per-capita growth less favorable at low abundance. A strong effect permits a stable zero state and a stable positive state separated by a finite threshold. Cooperative infections can exhibit the same positive-feedback structure: neither disease invades alone, but sufficiently many joint infections sustain both.

A food web records resource-consumer interactions among species. A linear food chain is a special case. Positive interaction coefficients alone do not imply persistence: the per-capita growth balances must admit feasible positive long-term mean abundances.

Two consumers with different break-even requirements cannot both have bounded persistent abundance on a single well-mixed limiting resource when their per-capita growth is linear in that resource. For $\dot{Y}/Y=cX-d$ and $\dot{Z}/Z=eX-f$, differentiation of $\log Y/c-\log Z/e$ gives the nonzero constant $-d/c+f/e$ when their break-even levels differ. This excludes simultaneous bounded persistence without assuming convergence to an equilibrium. Additional resources, environmental variation or different interactions can invalidate the simple one-resource argument.

A food chain orders species so that each consumer feeds on the level below and may be eaten by the level above. Basal resource growth, consumer mortality, conversion efficiency and density regulation determine whether all levels can persist.

For an unregulated exponential basal resource and nearest-neighbor Lotka-Volterra equations, the weighted interaction matrix is skew-symmetric matrix. An even chain has an invertible tridiagonal interaction matrix; its positive equilibrium, when feasible, fixes all long-term means. An odd chain has a null direction and requires a special balance between resource growth and mortality to permit bounded full coexistence. For a three-level chain, $X_1^{b_2}{X}_3^{a_1}=C\exp [(b_{2}r-a_1{d}_3)t]$. A positive even-chain equilibrium gives a conserved coercive relative-entropy-type function, proving every species remains bounded above and away from zero.

A network describing how gene products affect the production of other gene products. Positive and negative regulatory interactions can generate multistability, oscillations, and noise-sensitive transitions.

// Target: mathematical-biology.bigb

A repressilator is a cyclic negative-feedback gene regulatory network in which three repressors inhibit one another in sequence. In a normalized logic approximation for gene regulation, $x^{\prime }=H(h-z)-x$, $y^{\prime }=H(h-x)-y$, $z^{\prime }=H(h-y)-z$. For $0<h<1$, threshold crossings successively switch production and can generate sustained oscillations. A rise from $v_0<h$ reaches threshold after $\ln [(1-v_0)/(1-h)]$; a decay from $v_0>h$ reaches it after $\ln (v_0/h)$ in units of the common decay time.

A gene product reduces its own production. This negative feedback can stabilize its abundance and shorten the approach to a regulated level. In the strong logic approximation for gene regulation, production switches off near a self-repression threshold; a smooth repression law avoids artificial infinitely rapid switching at that threshold.

At a Hill equation equilibrium with $s=(Y/K{)}^n$, differentiating $\beta =\alpha Y(1+s)$ gives the displayed logarithmic sensitivity. It approaches one without effective feedback and $1/(n+1)$ under strong repression. Threshold pinning in the logic approximation for gene regulation instead gives approximately zero production sensitivity while the threshold is held fixed.

Neglecting the one in a strongly repressed Hill equation gives $\dot{Y}=\beta K^n{Y}^{-n}-\alpha Y$. The substitution $z=Y^{n+1}$ makes this a linear ordinary differential equation. It yields $Y_{\mathrm{st}}=(\beta K^n/\alpha {)}^{1/(n+1)}$ and half-response time $-\log (1-2^{-(n+1)})/[(n+1)\alpha ]$. The singular initial slope is an artifact of the approximation, whose range is not uniform at arbitrarily large $n$ and fixed production/degradation ratio.

A threshold approximation in which a regulatory input permits or blocks production according to whether a transcription factor exceeds a specified concentration. With first-order removal, $\dot{y}=\beta g-\alpha y$ and $g\in \{0,1\}$. A zero-initialized rising response reaches $K<\beta /\alpha$ at $-{\alpha }^{-1}\ln (1-\alpha K/\beta )$. Boolean rules must specify how several inputs combine; equal rate constants do not determine thresholds.

Suppose an input is produced for time $T$, then decays, and activates two identical mutually activating OR-gated outputs. Write $S=\beta /\alpha$ and $h=K/S\in (0,1)$. The input remains above threshold until ${\alpha }^{-1}\ln [(e^{\alpha T}-1)/h]$. Each output reaches threshold at $-2\ln (1-h)/\alpha$ if input remains available. Equating these times gives the limiting pulse duration $T_{\min }={\alpha }^{-1}\ln [1+h/(1-h{)}^2]$. Exact equality depends on the ideal gate's boundary convention; a robust latch requires a longer pulse. Treating cessation of production as immediate removal incorrectly discards the input's residual lifetime.

A feed-forward loop has regulatory paths $X\to Z$ and $X\to Y\to Z$. The two inputs are combined at the target gene using a specified regulatory rule.

In an incoherent feed-forward loop, the direct path from an input to an output and the indirect path through a second transcription factor have opposite net signs. Their different response times can generate a pulse or adaptation. A nonzero basal production rate makes the final adapted output nonzero, so its logic and response time must be calculated from the actual gene regulation rule rather than assumed from a Boolean label.

In the I4 incoherent feed-forward loop, an input activates the output but represses an intermediate activator of that output. Decay of the intermediate can interrupt an initial output pulse. Residual production and signal dependence must be retained when specifying the gene regulation rule.

In the I1 incoherent feed-forward loop, an input activates both an intermediate regulator and the output, while the intermediate represses the output. Delayed accumulation of the intermediate can therefore interrupt an initial output pulse. The precise steady state depends on the actual gene regulation rule and thresholds.

Let an output start at zero, degrade at rate $\alpha$, and switch from production $F\beta$ to $\beta$ at time $d$, with $F>1$. Its final level is $Z_s=\beta /\alpha$. Writing $\delta =1-e^{-\alpha d}$, its first crossing of $Z_s/2$ occurs at $t_{1/2}={\alpha }^{-1}\ln [2F/(2F-1)]$ if $F\delta \ge 1/2$; otherwise it occurs at $d+{\alpha }^{-1}\ln [2(1-F\delta )]$. This follows by solving the two linear differential equations before and after the switch. A large F alone does not make the first formula uniform in arbitrarily short pulse durations.

The direct path and the indirect path of a coherent feed-forward loop have matching overall signs. The all-activating case with an AND rule can delay activation while allowing rapid removal of the production signal when the direct input disappears.

The all-activating feed-forward loop combines direct activation of an output with indirect activation through an intermediate. Under AND logic the intermediate's accumulation delays production and can reject brief inputs. Removal of direct activation stops production immediately, while the output concentration still has its own decay time.

A persistence detector responds strongly to a sustained input and weakly to short isolated inputs. In an activating coherent feed-forward loop, an intermediate protein accumulates toward a threshold while the target also requires the direct input. Eliminating that intermediate by an instantaneous quasi-steady-state approximation removes its accumulation delay.

The interacting regulatory system built from pair-rule genes. Its interactions can change with developmental time rather than forming one static circuit.

An oscillatory developmental regulatory system associated with repeated segment formation. Oscillation, intercellular synchronisation and fixation of boundaries are distinct processes.

Two components mutually repress when each inhibits the other's production. With decreasing $f,g$, the two negative interactions create a positive feedback loop. Sufficiently strong repression can produce two stable states: high $x$, low $y$, and the reverse.

// Target: mathematical-biology.bigb

For $\dot{Y}=-Y+\gamma /(1+Z^2)$ and $\dot{Z}=-Z+\gamma /(1+Y^2)$, subtraction of the equilibrium conditions gives $(Y-Z)(1-YZ)=0$. The symmetric state solves $x^3+x=\gamma$. Two asymmetric positive states exist precisely for $\gamma >2$: $Y_{\pm }=(\gamma \pm \sqrt{{\gamma }^2-4})/2$, $Z_{\pm }=Y_{\mp }$. Their Jacobian matrix has eigenvalues $-1\pm 2/\gamma$, both negative; the symmetric state is then a saddle point. For $0<\gamma <2$ only the attracting symmetric state remains.

For $\dot{x}=\lambda /(1+y^m)-x$, $\dot{y}=\lambda /(1+x^n)-y$, the equilibrium slope product is $K=mn{x}^n{y}^m/[(1+x^n)(1+y^m)]$. A value $K>1$ gives a saddle separating attracting alternatives. Both exponents greater than one allow this at sufficiently large common $\lambda$. If both are at most one it is impossible. If $0<m\le 1<n$, define $Y(x)$ as the unique positive solution of $Y^{-1}+Y^{m-1}=x^{-1}+x^{n-1}$ and set $K_{m,n}(x)=mn{x}^{n}Y(x{)}^m/[(1+x^n)(1+Y(x{)}^m)]$. All equilibria are parameterized by $\lambda =x(1+Y^m)$, whose derivative is $(1+Y^m)(1-K)/(1-m{Y}^m/(1+Y^m))$. Thus multistability for some common strength occurs exactly when ${\max }_{x>0}{K}_{m,n}(x)>1$. For $m=1$ the maximum is $(n-1{)}^2[(n+1)/(n-1){]}^{(n+1)/n}/(4n)$. Consequently $mn>1$ alone is not sufficient when the two synthesis strengths are required to coincide.

For positive $m,n$, this mutual repression system can display bistability for some $\lambda$ exactly when both $m>1$ and $n>1$. If either exponent is at most one, it has a unique stable equilibrium for every $\lambda$. To prove the less obvious case $m\le 1<n$, put $a=x/(1+x)$ and $b=y/(1+y)$ at equilibrium. Equality of the two expressions for $\lambda$ gives $a\le b(1-b{)}^{n-1}$. The loop gain is then bounded by $mn{b}^2(1-b{)}^{n-1}$, whose maximum is $4mn(n-1{)}^{n-1}/(n+1{)}^{n+1}<1$. Swap the two components for the other case. If both exponents exceed one, sufficiently large $\lambda$ yields two stable equilibria with $(x,y)\sim (\lambda ,{\lambda }^{1-m})$ and $(x,y)\sim ({\lambda }^{1-n},\lambda )$. The weaker condition $mn\le 1$ is sufficient to exclude multistability, but does not exhaust all exclusions when the same production prefactor is imposed.

// Target: mathematical-biology.bigb

At an equilibrium $x_{\ast }=f(y_{\ast })$, $y_{\ast }=g(x_{\ast })$, the Jacobian matrix has eigenvalues $-1\pm \sqrt{f^{\prime }(y_{\ast })g^{\prime }(x_{\ast })}$. Its product of repression gains is also $(\partial \log f/\partial \log y)(\partial \log g/\partial \log x)$. Product below one gives a stable node, above one a saddle, and equality the threshold for a zero eigenvalue. Robust multiple stable equilibria require an intervening unstable equilibrium; a gain exceeding one alone is not a sufficient global criterion for a particular pair of repression functions.

// Target: mathematical-biology.bigb

The Monod equation is an empirical saturating law for the specific growth rate of a culture limited by one nutrient. Write $s\ge 0$ for its residual nutrient concentration, ${\mu }_{\max }>0$ for the limiting specific growth rate, and $K_s>0$ for the nutrient concentration at half that rate:Here $\mu$ and ${\mu }_{\max }$ have units of inverse time, whereas $K_s$ has the same units as $s$. The notation $K(C)$ in the title denotes the same growth function, with $C=s$, $K_{\max }={\mu }_{\max }$ and $K_0=K_s$. For $s\ll K_s$, $\mu (s)\approx ({\mu }_{\max }/K_s)s$; for $s\gg K_s$, $\mu (s)$ approaches ${\mu }_{\max }$. Its derivative ${\mu }^{\prime }(s)={\mu }_{\max }{K}_s/(K_s+s{)}^2$ is positive: more nutrient increases the specific growth rate, with progressively smaller increments.