# Mathematical Modeling and Analysis of Population Collapse

(a) The environmental impact (I) is the sum of population size (P) times per capita affluence level (A) times the impact of technologies (T). This is the “IPAT” equation: I = P x A x T. Non-linear effects, including thresholds and feedbacks, can amplify the environmental impact of human numbers, writes Harte. For example, a species may depend on a certain amount of intact habitat to survive. As human settlements encroach, a threshold is eventually crossed, and the species will, sometimes quite suddenly (within a generation or two), collapse. Feedback loops can also fast-forward environmental damage. A classic example is the loss of “albedo”: on a warming planet, there is less ice and snow to reflect heat back to space, so more sunlight is absorbed by the Earth’s surface, which intensifies warming. Harte observes that many feedbacks are fueled by population dynamics. For example, warming accelerates the decomposition of organic matter in cultivated soil. That decomposition, in turn, releases carbon dioxide into the atmosphere, which speeds even more warming. Because more people generally means more cultivated land, population growth affects the intensity of this feedback effect. students seek assistance through Environmental Studies Dissertation Help to get all their issues navigated and they can excel in this field of study. They provide the best feedback to them.

(b) Models of population dynamics (hereafter, referred to as population models) are useful tools for understanding, explaining, and predicting the dynamics and persistence of biological populations. From a management perspective, such models can be used for assessing the status of a population, diagnosing causes of population declines or explosive growth, prescribing management targets, and evaluating the prognosis of a population’s likely responses to alternative management actions Models of population dynamics can also help to predict populations’ responses to environmental changes, such as global climate change. Global climate change is predicted to influence arctic sea ice adversely, and this could affect the population dynamics and persistence of species that depend on sea ice environments. For example, polar bears depend on arctic sea ice for feeding and breeding. By integrating field data, climate-change models, and population models, Literature survey predicted that the polar bear population in the southern Beaufort Sea would experience a drastic decline because of a reduction in sea ice extent by the end of the 21st century.

(c) Research on EWS for tipping points in Earth Systems has developed toolkits to forecast change. These data-driven approaches seek to define EWS in time series data for critical transitions. The set of EWS criteria includes:

• A “critical slowing down,” where the response time to an external perturbation takes longer and longer to return to the steady state, the tipping point being when it ultimately never returns

• Increasing autocorrelation within the time series, as a correlation between subsequent states

• A skewing of the distribution of data when the time axis is collapsed

• A “flickering” between discrete values of the system variable as the system approaches a bifurcation

In simplified terms, the EWS toolkit can be illustrated in terms of the time t for a perturbed variable x to return to equilibrium :

dxdt=γ(x−a)(x−b).

This has two equilibria of opposing stability, one at x = a and another at x = b (here γ is a positive factor). The potentials are smooth, so the characteristic rate-of-return to equilibrium “−” (or departure “+”) is λ± = ±γ(a − b). As depicted in Figure 1A, this “potential well” analogy implies that a system approaching a tipping point may spend more time in the direction of the alternative attractor it is going to switch to than in the opposite direction, where the potential is steeper. This yields in skewing of the statistical distribution of the data, and also critical slowing down as the system approaches a bifurcation a = b, i.e., as λ → 0 and the return-time diverges (1/λ → ∞). “Flickering” between multiple attractors can occur where variability is high (either internally, or through the noise in the amplitude of the forcing mechanism), which may be why variance can perform better than the first order autocorrelation coefficient as an EWS

(d) Concern over climate change is at least as much a social network phenomenon as it is a rational decision based upon the existing evidence. For this reason, a way forward may be through mapping the parameter space of potential social cascades, rather than attempting to predict them. This “ecological” approach focuses on the connectivity and heterogeneity among systems of discrete, interacting agents, each with their own individual thresholds for conditions at which they are likely to change. In the application to a financial system, for example, banks are modeled as the agents, with inter-bank loans as their interconnections, and cash reserves as the threshold to each bank's failure. The heterogeneity among the agents in how they learn and respond to information is key to the stability of the financial system.

(e) Various types of tipping processes can be differentiated in the literature. Many authors refer to critical thresholds, a notion closely related to the metaphor of a “butterfly effect”. Other processes related to tipping dynamics include metamorphosis, where a rapid loss of structures of one sort occurs simultaneously with the development of new structures, as well as cascades driven by positive feedbacks in processes occurring simultaneously at smaller scales. The social tipping dynamics of interest for this study are typically manifested as spreading processes in complex social networks of behaviors, opinions, knowledge, technologies, and social norms including spreading processes of structural change and reorganization. These spreading processes resemble contagious dynamics observed in epidemiology that spread through social networks. Once triggered, such processes can be irreversible and difficult to stop. Similar contagious dynamics have been observed in human behavior , for example in assaultive violence , participation in social movements or health-related behaviors and traits, such as smoking or obesity. There is a need to understand STEs as functional subsystems of the planetary-scale World–Earth system consisting of interacting biophysical subsystems of the Earth, and the social, cultural, economic, and technological subsystems of the world of human societies. Potential STEs share one defining characteristic: A small change or intervention in the subsystem can lead to large changes at the macroscopic level and drive the World–Earth system into a new basin of attraction, making the transition difficult to reverse . Exact quantifications of the relationship between big and small are, however, rare, as are empirical examples. For the combination of big interventions and big effects, there are currently no convincing examples; however, the potential use solar radiation management geoengineering in the future would fall into this category. Finally, some changes in the World–Earth system might be driven by nonhuman and unintentional forces (e.g., a sufficiently large meteorite hitting the Earth or a disease outbreak), while others might be driven by conscious interventions of human agency.

(f) Tipping processes might be analyzed as a function of change in a suitably selected forcing variable or control parameter. The pertinent World–Earth system features such as the anthropogenic carbon emissions are commonly the product of complex interactions of multiple drivers. These factor can, however, in some cases be combined into a single dominant control parameter. In this study, the researchers identify a subsystem of the World–Earth system as a STE relevant for decarbonization transformation if it fulfils the following criteria:

C1. A set of parameters or drivers controlling its state can be described by a combined control parameter that after crossing a critical threshold (the STP) by a small amount influences a crucial system feature of relevance (here the rate of anthropogenic greenhouse gas emissions) leading to a qualitative change in the system after a reference time has passed allowing for the emergence of the effect.

C2. It is possible to differentiate particular human interventions leading to the small change in the control parameter that has a big effect on the crucial system feature, which will be referred to as the STI.

(g) Some researchers consider a mathematical model for the evolution and collapse of the Easter Island society. Based on historical reports, the available primary resources consisted almost exclusively in the trees, then we describe the inhabitants and the resources as an isolated dynamical system. A mathematical, and numerical, analysis about the Easter Island community collapse is performed. In particular, there is a need to analyze the critical values of the fundamental parameters and a demographic curve is presented. The technological parameter, quantifying the exploitation of the resources, is calculated and applied to the case of another extinguished civilization (Copán Maya) confirming the consistency of the adopted model.

(h) The study of the dynamics of populations is a tool of fundamental importance in this area, notably in Genetics, Ecology, and Epidemiology, to name just a few. In general, deterministic models in this field concern global or averaged features of the population, typically the size of certain sub-populations, or the proportion of individuals sharing certain characteristics. That is, the features of the population are averaged and the model aims at depicting the evolution of those averaged quantities as time passes. They are based on the implicit assumption that, roughly speaking, all individuals in a given sub-population behave essentially the same. Dynamics are usually modeled in discrete times through some difference equations, and through differential equations in continuous times. In turn, stochastic models are built either by adding noise terms to deterministic evolution equations, in order to take random fluctuations into account, or more interestingly, by considering individual behaviors which are then viewed as stochastic processes. Individual-based models permit in particular to consider how individuals collaborate or compete with each other for resources, or interact with their environment. Stochastic models of population dynamics rely essentially on Markov chains in discrete times, Markov jump processes and stochastic differential equations in continuous times, including notably branching processes and coalescent processes. The mechanisms driving dynamics of populations in nature are extremely intricate and involve a number of diverse features, whereas mathematical models must remain tractable and thus can only incorporate a few of them. In general, mathematical models for highly complex phenomena focus on a few key variables, and view the effects of the remaining ones as small (possibly random) perturbations of the simpler model. In this respect, deciding whether to opt for a deterministic versus a stochastic model may be a delicate issue. Deterministic models are simpler to solve analytically or numerically; random models can be considerably more complicated, in particular in the individual-based case, but it is generally admitted that they may be also more realist. One may wonder whether it is useful to handle more complex random models when a deterministic answer is expected anyway, and at the opposite, one may be concerned with the risk of missing some important consequences of randomness by making an oversimplified deterministic analysis. Of course, a first key question is whether a given mathematical model accurately describes a phenomenon of interest, which is usually answered by checking the agreement with experimental measurements. Once the scope of a model has been validated and the model is applied in concrete situations, another fundamental problem for practitioners is the comparison with available data in order to estimate its parameters, and then to be able to make reliable predictions about the future (or inferences about the past) of the population.

(i) Population is the most vital element of world but population projection has become one of the most serious problems in the world. Population sizes and growth in a country directly influence the situation of the economy, policy, culture, education and environment of that country and determine exploring the cost of natural resources. Every government and collective sectors always require proper idea about the future demands and consumptions for their future activities ,size of various subsistence like population, resources. To obtain this information, the behavior of the connected variables is analyzed based on the previous data by the statisticians and mathematicians and using the conclusions drawn from the analysis, they make future projections of the aimed at variable. Mathematical modeling is a broad interdisciplinary science that uses mathematical and computational techniques to model and elucidate the phenomena arising in real life problems. Thus, it is a process of mimicking reality by using the language of mathematics in terms of differential equations which describe the changing phenomena of the underlying systems. The population models determine the present state in terms of the past and the future state in terms of its present state which describes a typical human way of coping with the reality. The main reason for solving many differential equations is to learn the behavior about an underlying physical process that the equation is believed to model. Mathematical models can take many forms including dynamical systems, statistical models and differential equations. These and other types of models can overlap, with a given model involving a variety of abstract structures. A population model is a type of mathematical model that is applied to the study of population dynamics. Models allow a better understanding of how complex interactions and processes work. Modeling of dynamic interactions in nature can provide a manageable way of understanding how numbers change over time or in relation to each other.

### Models required for mathematical analysis:

Exponential growth and branching processes

Manthusian growth model

Logistic model

Model of Smith

Method of least square

Finite difference solution

Fish hatchery problem

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