Biology

The registered systems of skcomplex.systems.biology. The page of each shows its factory, its component types with their states, parameters, and laws as written, and, for the forms that have them, its equations as SymbolicAnalysis returns them, so it serves as the pattern for a system of your own.

Collective behaviour

• Active walkers and trail formation: Active walkers who walk between the corners of a green and wear a system of trails into it.
• Ant colony optimization on a road network: Ants that find the shortest path between two points of a Delaunay road network, round after round.
• Binary ant decision: Ants that choose between two equal branches to a food source, following the marks of earlier ants.
• Bird swarm with pair potentials: A swarm of self-propelled birds held together by a pair potential that attracts far and repels near.
• Boids in three dimensions: Boids in three dimensions, birds that flock by local rules in a periodic cube.
• Boids in two dimensions: Reynolds' boids in two dimensions, birds that flock by local rules without a leader.
• Calm and nervous rats: A colony of calm and nervous rats, in which nervousness spreads by contact and fades on its own.
• Couzin shoal with repulsion, alignment, and informed leaders: The Couzin shoal, in which a small informed minority leads a group that does not know who is informed.
• Pheromone foraging: Ants that find food and lay a pheromone trail home, which other ants follow: stigmergy.
• Strömbom shepherding of a flock: A shepherd that collects a scattered flock and drives it to a goal, the heuristic of Strömbom et al.

Ecology

• Agent-based Lotka-Volterra: Prey and predators that walk on a grid, whose numbers cycle like the Lotka-Volterra equations.
• COMETS: dynamic flux balance on a lattice with diffusion: An expanding colony of the E. coli core system on a lattice, which needs the extra metabolism.
• Carrying capacity of a meadow: Cows that graze a summer of 25 weeks on a meadow, which shows how many cows the meadow can carry.
• Competition: Two species that compete for the same resources, which coexist or exclude each other.
• Consumer-resource models of substitutable and essential resources: Consumer species that compete only through the resources they take up, by MacArthur's niche theory.
• Droop cell-quota growth: The Droop system, a phytoplankton whose growth rate follows the nutrient stored in its cells.
• Dynamic multi-species metabolic system: Two E. coli core strains that differ in maximal glucose uptake on one medium, which needs the extra.
• Exponential growth: A population that grows or declines at a constant per-capita rate.
• Fibonacci rabbits: Fibonacci's rabbits, N_{t+1} = N_t + N_{t-1}, where every pair older than one month breeds a new pair.
• Fisher-KPP: The Fisher-KPP equation, whose step invades the empty line as a front of speed 2 sqrt(r D).
• Food webs: A food web where basal species grow logistically and consumers live on Lotka-Volterra predation.
• Generalized Lotka-Volterra microbial community: The generalized Lotka-Volterra system of a microbial community, dx_i/dt = x_i (b_i + sum_j a_ij x_j).
• Generalized stochastic predator-prey: Prey and predators born and killed one at a time, with crowding of the prey and saturating predation.
• Hutchinson: Hutchinson's delayed logistic equation, which oscillates around K when r * tau exceeds π/2.
• Intrinsic-instability populations: Jensen's coupled populations, whose steady state becomes unstable without a change of the surroundings.
• Joint, community, and SteadyCom flux balance: Two members of the E. coli core system that share a medium, which needs the extra metabolism.
• Logistic growth: A population that grows logistically towards its carrying capacity.
• Logistic growth with a varying capacity: Logistic growth under a seasonal carrying capacity, which settles on a cycle of the season's period.
• Lotka-Volterra: Prey and predators whose populations oscillate through predation.
• Lotka-Volterra with harvesting: Lotka-Volterra predation where both species are harvested with a constant effort.
• Mangrove succession: The regrowth of a mangrove forest of the Sundarbans after a clear cut, from pioneers to climax trees.
• Mutualism: Two species that grow logistically and feed each other, above their carrying capacities.
• Neutral Moran system with speciation: A neutral community of constant size, where species arise by speciation and vanish by drift.
• Neutral birth-death with hopping between boxes: Organisms that divide, die, and hop between patches on a periodic lattice, and gather in clusters.
• Neutral macroecology: Many species under neutral birth, death, and immigration, whose abundances follow a gamma-like law.
• Nicholson-Bailey: Hosts and parasitoids in discrete generations, whose equilibrium is unstable to growing oscillations.
• Penaeid shrimp growth: The growth of one penaeid shrimp from the juvenile stage, forced by the food and the water temperature.
• Random Lotka-Volterra with higher-order interactions: A random Lotka-Volterra community with third-order interactions.
• Ricker map of overcompensating generations: Ricker's map of a population in discrete generations, n' = r n exp(-n), chaotic for large r.
• Rosenzweig-MacArthur and rate tipping: Fast prey and slow predators whose carrying capacity shrinks at a rate, which tips them when it is fast.
• Sequentially linear ecosystem: An ecosystem whose abundances follow linear dynamics between extinctions that rewrite its network.
• Size-structured forest with shading and seed dispersal: The McKendrick-von Foerster density of trees over position across a strip and height.
• Stigmergic predator-prey automaton: Plants, prey, and predators on a grid, whose animals share emotions through decaying marks.
• Wolf and sheep: The wolf and sheep predation system with one grass patch per cell.

Epidemiology

• Activity-driven temporal networks: SIS spreading on an activity-driven temporal network, whose edges last one step.
• Adaptive SIS with rewiring: The SIS epidemic on a network whose susceptible nodes cut their edges to infected neighbours.
• Agent-based SIR with mortality, waning immunity, incubation, and lockdown: People who walk on a grid and infect those in their cell, with optional extensions of the SIR system.
• Degree-class SIR on uncorrelated networks: The SIR epidemic on an uncorrelated network of power-law degrees, in heterogeneous mean field.
• Discrete-time SIRS with a recovery delay: Two cities with recurrent outbreaks of a disease of fixed infection and immunity periods.
• EpiSims city of people and locations: A synthetic city in the spirit of EpiSims, where people infect those they meet at home and at work.
• Heterogeneous mean-field SIS: The SIS epidemic on a network with a power-law degree distribution, in heterogeneous mean field.
• Metapopulation reaction-diffusion: SIR epidemics in subpopulations at the nodes of a network, linked by individuals who travel.
• Quenched mean-field SIS: The SIS epidemic on a fixed network in quenched mean field, one infection probability per node.
• Random-walker epidemics on a grid and on a small world: Random walkers among whom one sick walker starts a deadly epidemic, on a grid or on a network.
• Record-dynamics epidemic of resistant residents: Yearly strains that infect every resident less resistant than their strength, who become immune.
• SIR: An epidemic that moves individuals from susceptible to infected to recovered.
• SIR on a network as an agent system: SIR epidemics in cities at the nodes of a network, carried between cities by travellers.
• SIR with optimal vaccination: An SIR epidemic with the vaccination rate that minimises the burden of infections and vaccines.
• SIRS: An epidemic whose immunity wanes, which returns in damped waves to an endemic equilibrium.
• SIS: An infection without immunity, where the recovered become susceptible again and the disease is endemic.
• SIS on partnerships that dissolve and re-form: The SIS epidemic on partnerships that dissolve and are replaced at once, with k partners each.
• Watts-May SEIR with lasting partnerships: The SEIR epidemic of Watts and May, where partnerships that last longer spread the infection faster.
• Zombie outbreak of susceptible, zombie, and removed compartments: The outbreak of zombies of Munz et al., where the dead rise again and humans are eventually overrun.

Evolution

• Adaptive climbing in finite populations: A finite population that climbs a random fitness landscape by mutation and strong selection.
• Assembly growth along the assembly index: The numbers of objects at successive assembly indices, each discovered from the level below.
• Bak-Sneppen: The Bak-Sneppen system, whose fitnesses self-organise above a critical value in avalanches of extinction.
• Competitive Jain-Krishna model with signed couplings: The Jain-Krishna model with signed interactions, whose surviving species come to cooperate.
• Eigen quasispecies: The shares of a population at every number of mutations from the master sequence, the linear chain system.
• Genotype spaces and neutral networks of RNA folds: Walkers of RNA sequences that take one proposed point mutation per unit of time at temperature zero.
• Hypercycles: A network of self-replicating molecules whose catalysis keeps a set of them alive, a hypercycle.
• Jain-Krishna autocatalytic networks: The system of Jain and Krishna, where selection of the fittest builds autocatalytic sets of molecules.
• Kimura diffusion of allele frequencies: Replicate populations whose allele frequencies diffuse by drift until the allele is lost or fixed.
• Moran process: The Moran process, a population of constant size where fitter traits fix by selection and drift.
• NK fitness landscapes: Agents that climb Kauffman's NK fitness landscape from random genomes until they reach local peaks.
• NKCS coevolution on coupled NK landscapes: The NKCS model of Kauffman and Johnsen, species climbing NK landscapes coupled to each other's genes.
• Replicator dynamics and hawks and doves: The replicator dynamics of a game, by default hawks and doves, which settle at a share V / C of hawks.
• Rock-paper-scissors: Rock, paper, and scissors on a lattice, whose cyclic invasions keep the three strategies coexisting.
• Solé-Manrubia extinctions: The ecosystem of Solé and Manrubia, whose species go extinct when their dependencies turn against them.
• Strong-selection weak-mutation walk on fitness landscapes: A monomorphic population that climbs a fitness landscape as mutants arise and fix one at a time.
• Tangled Nature: The Tangled Nature system, where organisms whose types interact through random couplings coevolve.
• Undirected and directed polymer assembly pool: An assembly pool of linear polymers grown from one monomer by joining pairs of its polymers.

Neuroscience

• Adaptrode synapse: An Adaptrode synapse driven by a train of input pulses that stops after a while.
• Amari neural field: The Amari neural field, whose lateral inhibition holds a stationary bump of activity.
• Dense homogeneous associative network of transient attractors: The dense homogeneous associative network of Gros, whose memories become active one after the other.
• FitzHugh-Nagumo ODE: The FitzHugh-Nagumo neuron, the two-variable reduction of Hodgkin and Huxley's excitable membrane.
• Fukai-Tanaka winner-take-all network: A winner-take-all network of units with self-limited growth and mutual inhibition.
• Hodgkin-Huxley: The Hodgkin-Huxley neuron, whose sodium and potassium channels produce action potentials.
• Integrate-and-fire: The leaky integrate-and-fire neuron, tau v' = -(v - rest) + R I, which spikes and resets at a threshold.
• Izhikevich: The Izhikevich neuron, whose four parameters give regular spiking, bursting, chattering, and more.
• Kinouchi-Copelli excitable network: The Kinouchi-Copelli network of excitable elements, whose dynamic range peaks at the critical point.
• LEGION oscillators: A LEGION network, a lattice of relaxation oscillators that segments a stimulus into its objects.
• Networks of leaky integrate-and-fire neurons with delays: A random network of leaky integrate-and-fire neurons whose spikes reach their targets after a delay.
• Next-generation neural mass with theta forcing: The exact neural mass of quadratic integrate-and-fire neurons whose gamma rhythm nests in a theta drive.
• Phase units with activity-dependent conduction delays: Phase units coupled all to all through conduction delays that adapt to their activity, as myelination.
• Polyhomeostatic neuron: A neuron that adapts its gain and threshold so its firing rate approaches an exponential distribution.
• Theta neurons and QIF networks: A network of theta neurons, equivalently quadratic integrate-and-fire neurons, coupled all to all.
• Theta neurons with spike-timing-dependent plasticity: Theta neurons on a random directed network whose synapses learn by spike-timing-dependent plasticity.
• Wilson-Cowan: The Wilson-Cowan system of interacting excitatory and inhibitory populations of neurons.

Oncology

• Acquired and induced resistance with phenotype switching: Sensitive cells and cells resistant by mutation or by a reversible phenotype switch under a dose.
• Bone-marrow and blood cells with blasts and cancer stem cells: The bone marrow and blood system of leukaemia, in which blasts crowd out the normal cells.
• Cell-cycle compartments with killing, blocking, and recruiting agents: Tumour cells in cell-cycle compartments under killing, blocking, and recruiting agents.
• Core and periphery vasculature under anti-angiogenic therapy: A tumour with a live core, a live periphery, and a necrotic core, each fed by its own vasculature.
• Effector cells, tumour, and IL-2 under adoptive therapy: The Kirschner-Panetta system of effector cells, tumour cells, and IL-2 under adoptive therapy and IL-2.
• Exponential-linear tumour growth: A tumour that grows exponentially while small and linearly once its growth reaches a limiting rate.
• Generalized logistic (Richards) growth: A tumour that saturates at a carrying capacity, dn/dt = r n (1 - (n / K)^p), logistic for p = 1.
• Glioma, glia, and neurons under temozolomide and virus: Glia, neurons, and a glioma under a constant temozolomide infusion and an oncolytic virus.
• Gompertz growth: A tumour whose growth slows as the logarithm of the room left, dn/dt = r n log(K / n).
• Gompertz growth with a vascular carrying capacity: Gompertz growth under a carrying capacity that the vessels on the tumour surface keep raising.
• Gompertzian cell-cycle phases under tamoxifen: A breast tumour in Gompertzian cell-cycle phases under daily tamoxifen and its active metabolite.
• Hahnfeldt system with effector cells, pharmacokinetics, resistance, and toxicity: The Hahnfeldt tumour with effector cells under bevacizumab and atezolizumab every three weeks.
• Immune competition: The kinetic system of the competition between tumour cells and immune cells.
• Intermittent androgen suppression switched by PSA thresholds: The Ideta system of intermittent androgen suppression, switched on and off by PSA thresholds.
• Kirschner-Panetta system with TGF-beta and siRNA: The Kirschner-Panetta system with TGF-beta, which lets the tumour evade the immune system, and siRNA.
• Linear-quadratic cell survival and biologically effective dose: The linear-quadratic survival of cells after a total dose, with its biologically effective dose.
• Linear-quadratic impulsive irradiation with a radiosensitizer: A xenograft tumour irradiated in fractions scored by the linear-quadratic system, with a radiosensitizer.
• Multiscale tumour angiogenesis: A tumour that outgrows its oxygen supply and recruits new blood vessels from a parent vessel.
• Normal and leukaemic stem cells with feedback regulation: Normal and leukaemic stem cells that compete for one niche through the feedback of their progeny.
• Normal and malignant stem-cell compartments with feedback and therapy: The eight-compartment system of normal and malignant stem-cell lineages with feedback and therapy.
• Normal and tumour competition under radiation: Normal and tumour cells in Lotka-Volterra competition, the tumour irradiated periodically.
• Normal, tumour, and immune cells under chemotherapy: The de Pillis-Radunskaya system of immune, tumour, and normal cells under a constant drug infusion.
• Norton-Simon Gompertzian cell kill: The Norton-Simon system, a Gompertzian tumour killed in proportion to its size by a dose schedule.
• Oncolytic adenovirus with T cells and dendritic-cell injection: A tumour treated with an oncolytic adenovirus that expresses immune signals and with dendritic cells.
• Optimal adoptive therapy and IL-2 dosing: A tumour under the adoptive therapy and IL-2 injection that minimise its burden and their cost.
• Optimal chemotherapy dosing: A patient under the chemotherapy that minimises the tumour burden and the toxicity of the drug.
• Paclitaxel and trastuzumab with antibody binding: HER2+ breast cancer cells under trastuzumab and paclitaxel, with antibody binding and drug uptake.
• Piecewise-linear intermittent androgen suppression: The Hirata piecewise-linear system of intermittent androgen suppression for one of three patient types.
• Power-law tumour growth: A tumour whose proliferating part scales as a power of its size, dn/dt = r n^p.
• Predator-prey tumour-immune system of TCR gene therapy: The predator-prey system of a gene therapy that lets the T cells recognise the tumour.
• Primary and secondary sites with a migration delay under chemotherapy: Normal and tumour cells at a primary and a secondary site, with a delayed migration of tumour cells.
• Prostate cancer driven by activated androgen receptors: The Jain system of prostate epithelium and cancer driven by activated androgen receptors, with PSA.
• Prostate cancer under continuous androgen suppression: Prostate cancer under continuous androgen suppression, which androgen-independent cells survive.
• Prostate cancer with an androgen cell quota and PSA: The Portz cell-quota system of prostate cancer, whose cells grow on their internal androgen.
• Red-cell production under 6-mercaptopurine: The production of red cells under a constant level of 6-mercaptopurine, starting from health.
• Resistance subpopulations under three drugs: A tumour of eight populations resistant to subsets of three drugs, given by constant infusions.
• Resting and hunting immune cells with a conversion delay under chemotherapy: The tumour-immune system of resting and hunting cells with a conversion delay, under a drug infusion.
• Skipper-Schabel-Wilcox log-kill: The log-kill system, an exponentially growing tumour of which every dose kills a fixed fraction.
• T cells, tumour, and anti-PD-1 checkpoint inhibitor: Activated T cells, a tumour, and an anti-PD-1 drug that frees the T cells from checkpoint inhibition.
• T cells, tumour, dendritic cells, and IL-2 under dendritic-cell vaccination: Helper and cytotoxic T cells, a tumour, dendritic cells, and IL-2 under a dendritic-cell vaccine.
• Tumour and vascular volume under an angiogenesis inhibitor: The Hahnfeldt system, a tumour whose carrying capacity is the vasculature it stimulates and inhibits.
• Tumour, NK cells, CD8 T cells, and lymphocytes under chemoimmunotherapy: A tumour with NK cells, CD8 T cells, and lymphocytes under chemotherapy, IL-2, and TIL injection.
• Tumour-immune populations with division delays under dendritic-cell vaccination: A tumour and eight innate and adaptive immune populations, with division delays, under a vaccine.
• Tumour-immune system with dendritic licensing, IL-2, IL-10, and TGF-beta: A tumour against dendritic, cytotoxic, helper, and regulatory T cells with IL-2, IL-10, and TGF-beta.
• Two tumour clones under chemotherapy and dendritic-cell vaccine: Two competing tumour clones, sensitive and resistant to chemotherapy, under a drug and a vaccine.
• Uninfected and infected cells, virus, and chemotherapy: Uninfected and infected tumour cells, an oncolytic virus, and a chemotherapeutic drug.
• Uninfected and infected tumour cells under an oncolytic virus: Uninfected and infected tumour cells that share a carrying capacity, with infection by mass action.
• Von Bertalanffy growth: A tumour that grows by the balance of anabolism and catabolism, dn/dt = r n^p - d n.

Physiology

• Aliev-Panfilov cardiac tissue: The Aliev-Panfilov system of cardiac tissue, started from a broken wave that curls into a spiral.
• Barkley excitable medium with spirals and breakup: Barkley's excitable medium, started from orthogonal stripes whose crossing curls into a spiral wave.
• Blood glucose stabilised by antagonistic insulin and glucagon loops: Blood glucose stabilised by antagonistic hormone loops, insulin against highs and glucagon against lows.
• Cellular Potts model of cell sorting on square, triangular, and disordered lattices: The cellular Potts model of a confluent tissue whose cells of several kinds sort by differential adhesion.
• Coagulation cascade amplified by positive feedback and bounded by fibrinolysis: The coagulation cascade, an all-or-nothing burst of thrombin bounded by protein C and fibrinolysis.
• Cumulative drug toxicity: The toxicity a weekly drug accumulates in the body, which the metabolism clears once the doses stop.
• FitzHugh-Nagumo spirals: The FitzHugh-Nagumo excitable medium, started from a broken wave that curls into a spiral.
• Hyperbolic chemotaxis of cells and a chemical: Cells and a chemical moving with finite speed, whose two bumps of cells spread and attract each other.
• Intercellular exchange through gap junctions and tunnelling nanotubes on a cell network: The intercellular exchange of a substance through gap junctions and tunnelling nanotubes on a network.
• Lumped cardiovascular system: A closed lumped system of the systemic circulation driven by a left ventricle of varying elastance.
• Mackey-Glass: The Mackey-Glass equation x' = beta x(t - tau) / (1 + x(t - tau)^n) - gamma x of blood cell production.
• Steerable receptor-effector negative feedback loop: A receptor-effector loop that holds a product at its set point until a steering signal shifts it.
• Stimulus-response, adaptive response, and anticipation with response, restore, repair, and preparedness costs: A system that meets recurring demands by stimulus-response, adaptive response, or anticipation.
• Substance exchange between logistic cells in a closed contour: The exchange of signalling molecules between logistic cells arranged in a closed contour.
• Two-compartment pharmacokinetics: The two-compartment pharmacokinetics of a bolus dose given into the central compartment.
• Two-layer adaptive oscillator model of sepsis: The two-layer model of tumour disease and sepsis, parenchymal and immune cells as adaptive oscillators.
• Vessel conductances adapting to Kirchhoff flows: A square grid of vessels fed at one corner and drained at every other node, whose conductances adapt.
• West-Brown-Enquist distribution network and Kleiber scaling: The distribution network of West, Brown, and Enquist, a space-filling tree of invariant capillaries.

Systems biology

• Adenylate cyclase kinase cascade amplifier with G-protein timer inactivation: The adenylate cyclase cascade, which amplifies a hormone pulse and ends it by the G-protein timer.
• Boolean repressilator: The Boolean repressilator, where LacI represses TetR, TetR represses cI, and cI represses LacI.
• Boolean system of the mammalian cell cycle: The Boolean system of the mammalian cell cycle of Faure et al., ten proteins updated synchronously.
• Bristle patterning by lateral inhibition on a ring: A ring of cells that inhibit the neural fate of their neighbours, leaving sparse, evenly spaced bristles.
• Carbon-13 metabolic flux analysis: A metabolic network that LabellingAnalysis fits to labelling data, the TCA example by default.
• Drosophila circadian clock: Goldbeter's system of the Drosophila circadian clock, PER that represses its own transcription.
• Dynamic flux balance analysis of a batch culture: A batch culture of the E. coli core system, which needs the extra metabolism.
• Early-stage protein folding intermediate: The residues of a protein sampling the cells of an ellipse path of the Ramachandran map by heat bath.
• Elementary flux modes and extreme pathways: The toy network of Raman whose steady-state cone ElementaryModeAnalysis enumerates.
• Flux balance, flux variability, and parsimonious FBA: Flux balance analysis of a metabolic network, which sets its fluxes to maximise growth.
• Fuzzy oil drop system of protein structure: The fuzzy oil drop of a protein, an ensemble replayed one member per frame or a chain folded by moves.
• Generalised multi-valued logical networks: A generalised logical network of Thomas, whose variables step asynchronously towards focal levels.
• Generalized mass action and S-system power-law systems: A biochemical system of power-law rates, in the S-system or the generalized mass action form.
• Genetic toggle switch: The genetic toggle switch of Gardner, Cantor, and Collins, two genes that repress each other.
• Goodwin oscillator: The Goodwin oscillator, a chain of stages whose last one represses the synthesis of the first.
• Huepe-Aldana noisy majority Boolean network: The Boolean network of Huepe and Aldana-González, nodes that follow the noisy majority of k inputs.
• Kauffman random Boolean networks: A Kauffman random Boolean network, nodes updated together by random functions of k random inputs.
• Knock-outs, MoMA, ROOM, OptKnock, and synthetic lethals: The E. coli core system with some genes knocked out, which needs the extra metabolism.
• Min-protein skeleton system as mass-conserving reaction-diffusion: The skeleton system of the Min proteins on a periodic line, a mass-conserving reaction-diffusion system.
• Negative autoregulation and bursty gene expression: A gene that represses its own transcription and makes its protein in bursts.
• Probabilistic Boolean network: A probabilistic Boolean network, whose nodes draw one of their candidate functions at every step.
• Regulatory flux balance analysis: A batch culture on glucose and acetate with Boolean catabolite repression, which needs the extra.
• Repressilator: The repressilator of Elowitz and Leibler, three genes in a ring that each repress the next.
• Stochastic and continuous Petri nets: A Petri net, places that hold tokens and transitions that consume and produce them when they fire.
• Three-enzyme adaptation circuits of Ma et al.: A circuit of three enzymes A, B, and C of Ma et al., whose output C may adapt to a step of the input.
• Three-gene Boolean network: The three-gene Boolean network of Raman, updated synchronously.
• Three-node phosphorylation oscillator: Three proteins whose phosphorylation cycles regulate each other in a loop that oscillates.
• Three-way fate decision on the compactified elliptic umbilic: Cells that choose among three fates on the compactified elliptic umbilic landscape of Rand et al.
• Yeast cell-cycle network: The Boolean network of the budding yeast cell cycle of Li et al., eleven proteins and their regulation.
• Yeast glycolysis oscillations in a flow reactor: The glycolysis of a yeast extract in a flow reactor, whose NADH oscillates with a period of minutes.