In this paper we develop a compartmental model of SIR type (the abbreviation refers to the number of Susceptible, Infected and Recovered people) that models the population dynamics of two diseases that can coinfect. We discuss how the underlying dynamics depends on the carrying capacity K: from a simple dynamics to a more complex. This can also help in understanding the appearance of more complicated dynamics, for example, chaos and periodic oscillations, for large values of K. It is also presented that pathogens can invade in population and their invasion depends on the carrying capacity K which shows that the progression of disease in population depends on carrying capacity. More specifically, we establish all possible scenarios (the so-called transition diagrams) describing an evolution of an (always unique) locally stable equilibrium state (with only non-negative compartments) for fixed fundamental parameters (density independent transmission and vital rates) as a function of the carrying capacity K. An important implication of our results is the following important observation. Note that one can regard the value of K as the natural ‘size’ (the capacity) of a habitat. From this point of view, an isolation of individuals (the strategy which showed its efficiency for COVID-19 in various countries) into smaller resp. larger groups can be modelled by smaller resp. bigger values of K. Then we conclude that the infection dynamics becomes more complex for larger groups, as it fairly maybe expected for values of the reproduction number R0≈1. We show even more, that for the values R0>1 there are several (in fact four different) distinguished scenarios where the infection complexity (the number of nonzero infected classes) arises with growing K. Our approach is based on a bifurcation analysis which allows to generalize considerably the previous Lotka-Volterra model considered previously in Ghersheen et al. (Math Meth Appl Sci 42(8), 2019).

In this paper we continue the stability analysis of the model for coinfection with density dependent susceptible population introduced in Andersson et al. (Effect of density dependence on coinfection dynamics. arXiv:2008.09987, 2020). We consider the remaining parameter values left out from Andersson et al. (Effect of density dependence on coinfection dynamics. arXiv:2008.09987, 2020). We look for coexistence equilibrium points, their stability and dependence on the carrying capacity K. Two sets of parameter value are determined, each giving rise to different scenarios for the equilibrium branch parametrized by K. In both scenarios the branch includes coexistence points implying that both coinfection and single infection of both diseases can exist together in a stable state. There are no simple explicit expression for these equilibrium points and we will require a more delicate analysis of these points with a new bifurcation technique adapted to such epidemic related problems. The first scenario is described by the branch of stable equilibrium points which includes a continuum of coexistence points starting at a bifurcation equilibrium point with zero single infection strain #1 and finishing at another bifurcation point with zero single infection strain #2. In the second scenario the branch also includes a section of coexistence equilibrium points with the same type of starting point but the branch stays inside the positive cone after this. The coexistence equilibrium points are stable at the start of the section. It stays stable as long as the product of K and the rate γ¯γ¯ of coinfection resulting from two single infections is small but, after this it can reach a Hopf bifurcation and periodic orbits will appear.

We consider an age-structured density-dependent population model on several temporally variable patches. There are two key assumptions on which we base model setupand analysis. First, intraspecific competition is limited to competition between individuals of the same age (pure intra-cohort competition) and it affects density-dependentmortality. Second, dispersal between patches ensures that each patch can be reachedfrom every other patch, directly or through several intermediary patches, within individual reproductive age. Using strong monotonicity we prove existence and uniquenessof solution and analyze its large-time behavior in cases of constant, periodically variable and irregularly variable environment. In analogy to the next generation operator,we introduce the net reproductive operator and the basic reproduction number R0 fortime-independent and periodical models and establish the permanence dichotomy:if R₀ ≤ 1, extinction on all patches is imminent, and if R₀ > 1, permanence on allpatches is guaranteed. We show that a solution for the general time-dependent problemcan be bounded by above and below by solutions to the associated periodic problems. Using two-side estimates, we establish uniform boundedness and uniform persistenceof a solution for the general time-dependent problem and describe its asymptoticbehaviour

The population size has far-reaching effects on the fitness of the population, that, in its turn influences the population extinction or persistence. Understanding the density- and age-dependent factors will facilitate more accurate predictions about the population dynamics and its asymptotic behaviour. In this paper, we develop a rigourous mathematical analysis to study positive and negative effects of increased population density in the classical nonlinear age-structured population model introduced by Gurtin \& MacCamy in the late 1970s. One of our main results expresses the global stability of the system in terms of the newborn function only. We also derive the existence of a threshold population size implying the population extinction, which is well-known in population dynamics as an Allee effect.

Multiple viruses are widely studied because of their negative effect on the health of host as well as on whole population. The dynamics of coinfection are important in this case. We formulated an susceptible infected recovered (SIR) model that describes the coinfection of the two viral strains in a single host population with an addition of limited growth of susceptible in terms of carrying capacity. The model describes five classes of a population: susceptible, infected by first virus, infected by second virus, infected by both viruses, and completely immune class. We proved that for any set of parameter values, there exists a globally stable equilibrium point. This guarantees that the disease always persists in the population with a deeper connection between the intensity of infection and carrying capacity of population. Increase in resources in terms of carrying capacity promotes the risk of infection, which may lead to destabilization of the population.

We develop a novel approach to study the global behaviour of large foodwebs for ecosystems where several species share multiple resources. The model extends and generalizes some previous works and takes into account self-limitation. Under certain explicit conditions, we establish the global convergence and persistence of solutions.

An SIR model with the coinfection of the two infectious agents in a single host population is considered. The model includes the environmental carry capacity in each class of population. A special case of this model is analyzed, and several threshold conditions are obtained, which describes the establishment of diseases in the population. We prove that, for small carrying capacity K, there exists a globally stable disease-free equilibrium point. Furthermore, we establish the continuity of the transition dynamics of the stable equilibrium point, that is, we prove that, (1) for small values of K, there exists a unique globally stable equilibrium point, and (b) it moves continuously as K is growing (while its face type may change). This indicates that the carrying capacity is the crucial parameter and an increase in resources in terms of carrying capacity promotes the risk of infection.

We study the biodiversity problem for resource competition systems with extinctions and self-limitationeffects. Our main result establishes estimates of biodiversity in terms of the fundamental parameters ofthe model. We also prove the global stability of solutions for systems with extinctions and large turnoverrate. We show that when the extinction threshold is distinct from zero, the large time dynamics of systemis fundamentally non-predictable. In the last part of the paper we obtain explicit analytical estimates ofecosystem robustness with respect to variations of resource supply which support the R* rule for a systemwith random parameters.

Achieving food security will require closing yield gaps in many regions, including Pakistan. Although fertilizer subsidies have facilitated increased nitrogen (N) application rates, many staple crop yields have yet to reach their maximum potential. Considering that current animal manure and human excreta (bio-supply) recycling rates are low, there is substantial potential to increase the reuse of nutrients in bio-supply. We quantified 2010 crop N, phosphorus (P), and potassium (K) needs along with bio-supply nutrient availability for Pakistani districts, and compared these values to synthetic fertilizer use and costs. We found that synthetic fertilizer use combined with low bio-supply recycling resulted in a substantial gap between nutrient supply and P and K crop needs, which would cost 3 billion USD to fill with synthetic fertilizers. If all bio-supply was recycled, it could eliminate K synthetic fertilizer needs and decrease N synthetic fertilizer needs to 43% of what was purchased in 2010. Under a full recycling scenario, farmers would still require an additional 0.28 million tons of synthetic P fertilizers, costing 2.77 billion USD. However, it may not be prohibitively expensive to correct P deficiencies. Pakistan already spends this amount of money on fertilizers. If funds used for synthetic N were reallocated to synthetic P purchases in a full bio-supply recycling scenario, crop needs could be met. Most recycling could happen within districts, with only 6% of bio-supply requiring between-district transport when optimized to meet national N crop needs. Increased recycling in Pakistan could be a viable way to decrease yield gaps.