Mary Lou Zeeman

Hopf bifurcations in competitive three-dimensional Lotka-Volterra Systems

We study the space of Lotka–Volterra systems modelling three mutually competing species, each of which, in isolation, would exhibit logistic growth. By a theorem of M. W. Hirsch, the compact limit sets of these systems are either fixed points or periodic orbits. We use a geometric analysis of the surfaces ẋ=0 of a system, to define a combinatorial equivalence relation on the space, in terms of simple inequalities on the parameters. We list the 33 stable equivalence classes, and show that in 25 of these classes all the compact limit sets are fixed points, so we can fully describe the dynamics. We study the remaining eight equivalence classes by finding simple algebraic criteria on the parameters, with which we are able to predict the occurrence of Hopf bifurcations and, consequently, isolated periodic orbits.

Three-dimensional competitive Lotka-Volterra systems with no periodic orbits

The following conjecture of M. L. Zeeman is proved. If three interacting species modeled by a competitive Lotka--Volterra system can each resist invasion at carrying capacity, then there can be no coexistence of the species. Indeed, two of the species are driven to extinction. It is also proved that in the other extreme, if none of the species can resist invasion from either of the others, then there is stable coexistence of at least two of the species. In this case, if the system has a fixed point in the interior of the positive cone in R3 , then that fixed point is globally asymptotically stable, representing stable coexistence of all three species. Otherwise, there is a globally asymptotically stable fixed point in one of the coordinate planes of R3 , representing stable coexistence of two of the species.

Extinction in competitive Lotka-Volterra systems

It is well known that for the two species autonomous competitive Lotka-Volterra model with no fixed point in the open positive quadrant, one of the species is driven to extinction, whilst the other population stabilises at its own carrying capacity. In this paper we prove a generalisation of this result to arbitrary finite dimension. That is, for the n-species autonomous competitive Lotka-Volterra model, we exhibit simple algebraic criteria on the parameters which guarantee that all but one of the species is driven to extinction, whilst the one remaining population stabilises at its own carrying capacity.

From local to global behavior in competitive Lotka-Volterra systems

In this paper we exploit the linear, quadratic, monotone and geometric structures of competitive Lotka-Volterra systems of arbitrary dimension to give geometric, algebraic and computational hypotheses for ruling out nontrivial recurrence. We thus deduce the global dynamics of a system from its local dynamics. The geometric hypotheses rely on the introduction of a split Liapunov function. We show that if a system has a fixed point p ∈ int R n + and the carrying simplex of the system lies to one side of its tangent hyperplane at p, then there is no nontrivial recurrence, and the global dynamics are known. We translate the geometric hypotheses into algebraic hypotheses in terms of the definiteness of a certain quadratic function on the tangent hyperplane. Finally, we derive a computational algorithm for checking the algebraic hypotheses, and we compare this algorithm with the classical Volterra-Liapunov stability theorem for Lotka-Volterra systems.

An n-dimensional competitive Lotka–Volterra system is generically determined by the edges of its carrying simplex

In an n-dimensional competitive Lotka–Volterra system on Rn+ the carrying simplex Σ is the invariant (n−1)-dimensional surface, homeomorphic to the standard unit simplex, that attracts all non-zero orbits and carries the asymptotic dynamics (Hirsch M W 1988 Nonlinearity 1 51–71, Zeeman M L 1993 Dynam. Stab. Sys. 8 189–217). We show that the one-dimensional edges of Σ (as sets) generically determine the system up to multiplication by a scalar, and hence determine Σ and the phase portrait.

Disease Induced Oscillations between Two Competing Species

The interaction of disease and competition dynamics is investigated in a system of two competing species in which only one species is susceptible to disease. The model is kept as simple as possible, combining Lotka--Volterra competition between the species with disease dynamics of susceptible and infective individuals within one of the species. It is assumed that pure vertical disease transmission (from parent to offspring) dominates horizontal transmission (by contact between infective and susceptible individuals) and that infective individuals have the same competition strength as susceptibles but a lower intrinsic growth rate. These assumptions yield three-dimensional competitive Lotka--Volterra dynamics modeling the disease-competition interaction. It is proved that if in the absence of disease there is competitive exclusion between the two species, then the presence of disease can lead to stable or oscillatory coexistence of both species. The case of oscillatory coexistence can be viewed either as di...

Resonance in the menstrual cycle: a new model of the LH surge

In vertebrates, ovulation is triggered by a surge of LH from the pituitary. The precise mechanism by which rising oestradiol concentrations initiate the LH surge in the human menstrual cycle remains a fundamental open question of reproductive biology. It is well known that sampling of serum LH on a time scale of minutes reveals pulsatile release from the pituitary in response to pulses of gonadotrophin releasing hormone from the hypothalamus. The LH pulse frequency and amplitude vary considerably over the cycle, with the highest frequency and amplitude at the midcycle surge. Here a new mathematical model is presented of the pituitary as a damped oscillator (pulse generator) driven by the hypothalamus. The model LH surge is consistent with LH data on the time scales of both minutes and days. The model is used to explain the surprising pulse frequency characteristics required to treat human infertility disorders such as Kallmanns syndrome, and new experimental predictions are made.

Constant proportion harvest policies: Dynamic implications in the Pacific halibut and Atlantic cod fisheries

Overfishing, pollution and other environmental factors have greatly reduced commercially valuable stocks of fish. In a 2006 Science article, a group of ecologists and economists warned that the world may run out of seafood from natural stocks if overfishing continues at current rates. In this paper, we explore the interaction between a constant proportion harvest policy and recruitment dynamics. We examine the discrete-time constant proportion harvest policy discussed in Ang et al. (2009) and then expand the framework to include stock-recruitment functions that are compensatory and overcompensatory, both with and without the Allee effect. We focus on constant proportion policies (CPPs). CPPs have the potential to stabilize complex overcompensatory stock dynamics, with or without the Allee effect, provided the rates of harvest stay below a threshold. If that threshold is exceeded, CPPs are known to result in the sudden collapse of a fish stock when stock recruitment exhibits the Allee effect. In case studies, we analyze CPPs as they might be applied to Gulf of Alaska Pacific halibut fishery and the Georges Bank Atlantic cod fishery based on harvest rates from 1975 to 2007. The best fit models suggest that, under high fishing mortalities, the halibut fishery is vulnerable to sudden population collapse while the cod fishery is vulnerable to steady decline to zero. The models also suggest that CPP with mean harvesting levels from the last 30 years can be effective at preventing collapse in the halibut fishery, but these same policies would lead to steady decline to zero in the Atlantic cod fishery. We observe that the likelihood of collapse in both fisheries increases with increased stochasticity (for example, weather variability) as predicted by models of global climate change.

Pituitary network connectivity as a mechanism for the luteinising hormone surge.

Ovulation in vertebrates is caused by a surge of luteinising hormone (LH) from the pituitary. The LH surge is initiated by rising oestradiol concentration, although the precise mechanism of oestradiol action in humans and primates is not yet understood. Recent advances in labelling and three‐dimensional imaging have revealed a rich pituitary structure of interwoven networks of different cell types. In the present study, we develop a mathematical model to test the hypothesis that oestradiol modulation of connectivity between pituitary cells can underlie the LH surge. In the model, gonadotrophin‐releasing hormone (GnRH) pulses stimulate LH secretion by two independent mechanisms. The first mechanism corresponds to the well known direct action of GnRH on gonadotrophs, which is inhibited by the rising oestradiol concentration. The second mechanism of GnRH action is to stimulate a recurrent network of pituitary cells; in this case, the folliculostellate cells, which in turn stimulate LH secretion from the gonadotrophs. The network activity is modelled by a one‐dimensional ordinary differential equation. The key to the LH surge in the model lies in the assumption that oestradiol modulates network connectivity. When the circulating oestradiol concentration is low, the network is barely connected, and cannot maintain a recurrent signal. When the oestradiol concentration is high, the network is highly connected, and maintains a high level of activity even after GnRH stimulation, thereby leading to a surge of LH secretion.

Social stress alters expression of large conductance calcium-activated potassium channel subunits in mouse adrenal medulla and pituitary glands.

Large conductance calcium‐activated potassium (BK) channels are very prominently expressed in adrenal chromaffin and many anterior pituitary cells, where they shape intrinsic excitability complexly. Stress‐ and sex‐steroids regulate alternative splicing of Slo‐α, the pore‐forming subunit of BK channels, and chronic behavioural stress has been shown to alter Slo splicing in tree shrew adrenals. In the present study, we focus on mice, measuring the effects of chronic behavioural stress on total mRNA expression of the Slo‐α gene, two key BK channel β subunit genes (β2 and β4), and the ‘STREX’ splice variant of Slo‐α. As a chronic stressor, males of the relatively aggressive SJL strain were housed with a different unfamiliar SJL male every 24 h for 19 days. This ‘social‐instability’ paradigm stressed all individuals, as demonstrated by reduced weight gain and elevated corticosterone levels. Five quantitative reverse transcriptase‐polymerase chain assays were performed in parallel, including β‐actin, each calibrated against a dilution series of its corresponding cDNA template. Stress‐related changes in BK expression were larger in mice tested at 6 weeks than 9 weeks. In younger animals, Slo‐α mRNA levels were elevated 44% and 116% in the adrenal medulla and pituitary, respectively, compared to individually‐housed controls. β2 and β4 mRNAs were elevated 162% and 194% in the pituitary, but slightly reduced in the adrenals of stressed animals. In the pituitary, dominance scores of stressed animals correlated negatively with α and β subunit expression, with more subordinate individuals exhibiting levels that were three‐ to four‐fold higher than controls or dominant individuals. STREX variant representation was lower in the subordinate subset. Thus, the combination of subunits responding to stress differs markedly between adrenal and pituitary glands. These data suggest that early stress will differentially affect neuroendocrine cell excitability, and call for detailed analysis of functional consequences.