Predator-prey models with component Allee effect for predator reproduction.

We present four predator-prey models with component Allee effect for predator reproduction. Using numerical simulation results for our models, we describe how the customary definitions of component and demographic Allee effects, which work well for single species models, can be extended to predators in predator-prey models by assuming that the prey population is held fixed. We also find that when the prey population is not held fixed, then these customary definitions may lead to conceptual problems. After this discussion of definitions, we explore our four models, analytically and numerically. Each of our models has a fixed point that represents predator extinction, which is always locally stable. We prove that the predator will always die out either if the initial predator population is sufficiently small or if the initial prey population is sufficiently small. Through numerical simulations, we explore co-existence fixed points. In addition, we demonstrate, by simulation, the existence of a stable limit cycle in one of our models. Finally, we derive analytical conditions for a co-existence trapping region in three of our models, and show that the fourth model cannot possess a particular kind of co-existence trapping region. We punctuate our results with comments on their real-world implications; in particular, we mention the possibility of prey resurgence from mortality events, and the possibility of failure in a biological pest control program.

A minimal spatio-temporal model of the NF-κB signalling pathway exhibits a range of behaviours.

In animal cells, the transcription factor NF-κB regulates many stressful, inflammatory, and innate immune responses. Experiments have revealed that, in response to cell stimulation, NF-κB can exhibit oscillatory dynamics where the nature of these dynamics can influence the pattern of NF-κB-dependent gene expression. Oscillations in NF-κB are believed to depend on a negative feedback loop linking NF-κB and one of its downstream products, namely IκBα. This negative feedback loop is enhanced by cell stimulation. However, it also exists in the absence of cell stimulation. Here we propose a minimal spatio-temporal model of the NF-κB signalling pathway, composed of partial differential equations. Through numerical simulations, we find various combinations of behaviours before and during cell stimulation: equilibrium dynamics (rapid convergence to a solution that is everywhere constant) before cell stimulation, followed by oscillatory dynamics during cell stimulation; oscillatory dynamics before and during cell stimulation; oscillatory dynamics before cell stimulation, followed by equilibrium dynamics during cell stimulation; and equilibrium dynamics before and during cell stimulation. In each case, when cell stimulation ceases, the model quickly returns to its pre-stimulation behaviour. All of these different combinations of behaviours occur for similar sets of parameter values. Therefore, our results may help to explain why, in experiments on the NF-κB pathway involving populations of cells, only a certain fraction of the cells exhibit oscillatory dynamics.

Influence of the nuclear membrane, active transport, and cell shape on the Hes1 and p53-Mdm2 pathways: insights from spatio-temporal modelling.

There are many intracellular signalling pathways where the spatial distribution of the molecular species cannot be neglected. These pathways often contain negative feedback loops and can exhibit oscillatory dynamics in space and time. Two such pathways are those involving Hes1 and p53-Mdm2, both of which are implicated in cancer. In this paper we further develop the partial differential equation (PDE) models of Sturrock et al. (J. Theor. Biol., 273:15-31, 2011) which were used to study these dynamics. We extend these PDE models by including a nuclear membrane and active transport, assuming that proteins are convected in the cytoplasm towards the nucleus in order to model transport along microtubules. We also account for Mdm2 inhibition of p53 transcriptional activity. Through numerical simulations we find ranges of values for the model parameters such that sustained oscillatory dynamics occur, consistent with available experimental measurements. We also find that our model extensions act to broaden the parameter ranges that yield oscillations. Hence oscillatory behaviour is made more robust by the inclusion of both the nuclear membrane and active transport. In order to bridge the gap between in vivo and in silico experiments, we investigate more realistic cell geometries by using an imported image of a real cell as our computational domain. For the extended p53-Mdm2 model, we consider the effect of microtubule-disrupting drugs and proteasome inhibitor drugs, obtaining results that are in agreement with experimental studies.

Spatio-temporal modelling of the NF-κB intracellular signalling pathway: the roles of diffusion, active transport, and cell geometry.

The nuclear factor kappa B (NF-κB) intracellular signalling pathway is central to many stressful, inflammatory, and innate immune responses. NF-κB proteins themselves are transcription factors for hundreds of genes. Experiments have shown that the NF-κB pathway can exhibit oscillatory dynamics-a negative feedback loop causes oscillatory nuclear-cytoplasmic translocation of NF-κB. Given that cell size and shape are known to influence intracellular signal transduction, we consider a spatio-temporal model of partial differential equations for the NF-κB pathway, where we model molecular movement by diffusion and, for several key species including NF-κB, by active transport as well. Through numerical simulations we find values for model parameters such that sustained oscillatory dynamics occur. Our spatial profiles and animations bear a striking resemblance to experimental images and movie clips employing fluorescent fusion proteins. We discover that oscillations in nuclear NF-κB may occur when active transport is across the nuclear membrane only, or when no species are subject to active transport. However, when active transport is across the nuclear membrane and NF-κB is additionally actively transported through the cytoplasm, oscillations are lost. Hence transport mechanisms in a cell will influence its response to activation of its NF-κB pathway. We also demonstrate that sustained oscillations in nuclear NF-κB are somewhat robust to changes in the shape of the cell, or the shape, location, and size of its nucleus, or the location of ribosomes. Yet if the cell is particularly flat or the nucleus sufficiently small, then oscillations are lost. Thus the geometry of a cell may partly determine its response to NF-κB activation. The NF-κB pathway is known to be constitutively active in several human cancers. Our spatially explicit modelling approach will allow us, in future work, to investigate targeted drug therapy of tumours.

Spatio-temporal modelling of the Hes1 and p53-Mdm2 intracellular signalling pathways.

The correct localisation of transcription factors is vitally important for the proper functioning of many intracellular signalling pathways. Experimental data has shown that many pathways exhibit oscillations in concentrations of the substances involved, both temporally and spatially. Negative feedback loops are important components of these oscillations, providing fine regulation for the factors involved. In this paper we consider mathematical models of two such pathways-Hes1 and p53-Mdm2. Building on previous mathematical modelling approaches, we derive systems of partial differential equations to capture the evolution in space and time of the variables in the Hes1 and p53-Mdm2 systems. Through computational simulations we show that our reaction-diffusion models are able to produce sustained oscillations both spatially and temporally, accurately reflecting experimental evidence and advancing previous models. The simulations of our models also allow us to calculate a diffusion coefficient range for the variables in each mRNA and protein system, as well as ranges for other key parameters of the models, where sustained oscillations are observed. Finally, by exploiting the explicitly spatial nature of the partial differential equations, we are also able to manipulate mathematically the spatial location of the ribosomes, thus controlling where the proteins are synthesized within the cytoplasm. The results of these simulations predict an optimal distance outside the nucleus where protein synthesis should take place in order to generate sustained oscillations. Using partial differential equation models, new information can be gained about the precise spatio-temporal dynamics of mRNA and proteins. The ability to determine spatial localisation of proteins within the cell is likely to yield fresh insight into a range of cellular diseases such as diabetes and cancer.

Perverse consequences of infrequently culling a pest.

There are potentially many situations in which creatures will be subject to infrequent but regular culling. In terms of controlling crop pests, some farmers may only be able to afford to apply pesticides occasionally. Alternatively, pesticides may be applied only occasionally to limit their unwelcome side effects, which include pesticide resistance, chemical poisoning of agricultural workers, and environmental degradation. In terms of conservation, some species (such as the red deer in the UK) may be culled occasionally to maintain balances within their ecosystem. However, in this paper we discover, as the culmination of an exploration of adult-stage culling of a creature with juvenile and adult life stages, that, in certain circumstances, regular but infrequent culling will, perversely, increase the average population of the creature.

Impulsive culling of a structured population on two patches.

We consider a model for a creature inhabiting two patches between which migration may occur. The creature is assumed to have a life cycle with two stages, namely juvenile and adult, giving rise to a delay differential system. The creature could represent an insect crop pest whilst the patches could represent neighbouring farms. Given that it is common to control crop pests by adult impulsive culling, we impose an adult impulsive culling regime on each patch. We find conditions on the regimes such that the pest will be eradicated on both patches simultaneously. The regime on one patch is assumed to be independent of the regime on the other patch to reflect the possibility that the patches represent farms with different owners where each owner has autonomy in their pest control decisions. In the special case where the birth functions on both patches are of an Allee type, we calculate explicit finite upper bounds for the number of culls needed on each patch to guarantee eradication. Simulations corroborate our theoretical results.

Biocontrol in an impulsive predator-prey model.

We study a model for biological pest control (or "biocontrol") in which a pest population is controlled by a program of periodic releases of a fixed yield of predators that prey on the pest. Releases are represented as impulsive increases in the predator population. Between releases, predator-pest dynamics evolve according to a predator-prey model with some fairly general properties: the pest population grows logistically in the absence of predation; the predator functional response is either of Beddington-DeAngelis type or Holling type II; the predator per capita birth rate is bounded above by a constant multiple of the predator functional response; and the predator per capita death rate is allowed to be decreasing in the predator functional response and increasing in the predator population, though the special case in which it is constant is permitted too. We prove that, when the predator functional response is of Beddington-DeAngelis type and the predators are not sufficiently voracious, then the biocontrol program will fail to reduce the pest population below a particular economic threshold, regardless of the frequency or yield of the releases. We prove also that our model possesses a pest-eradication solution, which is both locally and globally stable provided that predators are sufficiently voracious and that releases occur sufficiently often. We establish, curiously, that the pest-eradication solution can be locally stable whilst not being globally stable, the upshot of which is that, if we delay a biocontrol response to a new pest invasion, then this can change the outcome of the response from pest eradication to pest persistence. Finally, we state a number of specific examples for our model, and, for one of these examples, we corroborate parts of our analysis by numerical simulations.

A predator-prey model with generic birth and death rates for the predator.

We propose and study a predator-prey model in which the predator has a Holling type II functional response and generic per capita birth and death rates. Given that prey consumption provides the energy for predator activity, and that the predator functional response represents the prey consumption rate per predator, we assume that the per capita birth and death rates for the predator are, respectively, increasing and decreasing functions of the predator functional response. These functions are monotonic, but not necessarily strictly monotonic, for all values of the argument. In particular, we allow the possibility that the predator birth rate is zero for all sufficiently small values of the predator functional response, reflecting the idea that a certain level of energy intake is needed before a predator can reproduce. Our analysis reveals that the model exhibits the behaviours typically found in predator-prey models - extinction of the predator population, convergence to a periodic orbit, or convergence to a co-existence fixed point. For a specific example, in which the predator birth and death rates are constant for all sufficiently small or large values of the predator functional response, we corroborate our analysis with numerical simulations. In the unlikely case where these birth and death rates equal the same constant for all sufficiently large values of the predator functional response, the model is capable of structurally unstable behaviour, with a small change in the initial conditions leading to a more pronounced change in the long-term dynamics.

A spatio-temporal model of Notch signalling in the zebrafish segmentation clock: conditions for synchronised oscillatory dynamics.

In the vertebrate embryo, tissue blocks called somites are laid down in head-to-tail succession, a process known as somitogenesis. Research into somitogenesis has been both experimental and mathematical. For zebrafish, there is experimental evidence for oscillatory gene expression in cells in the presomitic mesoderm (PSM) as well as evidence that Notch signalling synchronises the oscillations in neighbouring PSM cells. A biological mechanism has previously been proposed to explain these phenomena. Here we have converted this mechanism into a mathematical model of partial differential equations in which the nuclear and cytoplasmic diffusion of protein and mRNA molecules is explicitly considered. By performing simulations, we have found ranges of values for the model parameters (such as diffusion and degradation rates) that yield oscillatory dynamics within PSM cells and that enable Notch signalling to synchronise the oscillations in two touching cells. Our model contains a Hill coefficient that measures the co-operativity between two proteins (Her1, Her7) and three genes (her1, her7, deltaC) which they inhibit. This coefficient appears to be bounded below by the requirement for oscillations in individual cells and bounded above by the requirement for synchronisation. Consistent with experimental data and a previous spatially non-explicit mathematical model, we have found that signalling can increase the average level of Her1 protein. Biological pattern formation would be impossible without a certain robustness to variety in cell shape and size; our results possess such robustness. Our spatially-explicit modelling approach, together with new imaging technologies that can measure intracellular protein diffusion rates, is likely to yield significant new insight into somitogenesis and other biological processes.

Pulse vaccination strategies in a metapopulation SIR model.

We examine a model for a disease with SIR-type dynamics circulating in a population living on two or more patches between any pair of which migration is allowed. We suppose that a pulse vaccination strategy (PVS) is carried out on each patch. Conditions are derived on each PVS such that the disease will be eradicated on all patches. The PVS on one patch is assumed to be essentially independent of the PVS on the other patches except in so far as they are all performed simultaneously. This independence is of practical value when we bear in mind that the patches may represent regions or countries with autonomous public health authorities, which may make individual decisions about the days appropriate for a vaccination pulse to occur in their own region or country. Simulations corroborate our theoretical results.