DOI: 1101.0018Jan 4, 2011Paper

General Properties of a System of $S$ Species Competing Pairwise

R. K. P. Zia


We consider a system of $N$ individuals consisting of $S$ species that interact pairwise: $x_m+x_\ell \rightarrow 2x_m\,\,$ with arbitrary probabilities $p_m^\ell $. With no spatial structure, the master equation yields a simple set of rate equations in a mean field approximation, the focus of this note. Generalizing recent findings of cyclically competing three- and four-species models, we cast these equations in an appealingly simple form. As a result, many general properties of such systems are readily discovered, e.g., the major difference between even and odd $S$ cases. Further, we find the criteria for the existence of (subspaces of) fixed points and collective variables which evolve trivially (exponentially or invariant). These apparently distinct aspects can be traced to the null space associated with the interaction matrix, $p_m^\ell $. Related to the left- and right- zero-eigenvectors, these appear to be "dual" facets of the dynamics. We also remark on how the standard Lotka-Volterra equations (which include birth/death terms) can be regarded as a special limit of a pairwise interacting system.

Related Concepts

Trending Feeds


Coronaviruses encompass a large family of viruses that cause the common cold as well as more serious diseases, such as the ongoing outbreak of coronavirus disease 2019 (COVID-19; formally known as 2019-nCoV). Coronaviruses can spread from animals to humans; symptoms include fever, cough, shortness of breath, and breathing difficulties; in more severe cases, infection can lead to death. This feed covers recent research on COVID-19.

STING Receptor Agonists

Stimulator of IFN genes (STING) are a group of transmembrane proteins that are involved in the induction of type I interferon that is important in the innate immune response. The stimulation of STING has been an active area of research in the treatment of cancer and infectious diseases. Here is the latest research on STING receptor agonists.

Chronic Fatigue Syndrome

Chronic fatigue syndrome is a disease characterized by unexplained disabling fatigue; the pathology of which is incompletely understood. Discover the latest research on chronic fatigue syndrome here.

Hereditary Sensory Autonomic Neuropathy

Hereditary Sensory Autonomic Neuropathies are a group of inherited neurodegenerative disorders characterized clinically by loss of sensation and autonomic dysfunction. Here is the latest research on these neuropathies.

Spatio-Temporal Regulation of DNA Repair

DNA repair is a complex process regulated by several different classes of enzymes, including ligases, endonucleases, and polymerases. This feed focuses on the spatial and temporal regulation that accompanies DNA damage signaling and repair enzymes and processes.

Glut1 Deficiency

Glut1 deficiency, an autosomal dominant, genetic metabolic disorder associated with a deficiency of GLUT1, the protein that transports glucose across the blood brain barrier, is characterized by mental and motor developmental delays and infantile seizures. Follow the latest research on Glut1 deficiency with this feed.

Separation Anxiety

Separation anxiety is a type of anxiety disorder that involves excessive distress and anxiety with separation. This may include separation from places or people to which they have a strong emotional connection with. It often affects children more than adults. Here is the latest research on separation anxiety.

KIF1A Associated Neurological Disorder

KIF1A associated neurological disorder (KAND) is a rare neurodegenerative condition caused by mutations in the KIF1A gene. KAND may present with a wide range and severity of symptoms including stiff or weak leg muscles, low muscle tone, a lack of muscle coordination and balance, and intellectual disability. Find the latest research on KAND here.

Regulation of Vocal-Motor Plasticity

Dopaminergic projections to the basal ganglia and nucleus accumbens shape the learning and plasticity of motivated behaviors across species including the regulation of vocal-motor plasticity and performance in songbirds. Discover the latest research on the regulation of vocal-motor plasticity here.

Related Papers

Neural Networks : the Official Journal of the International Neural Network Society
Yves MoreauLeon Brenig
© 2021 Meta ULC. All rights reserved