Mathematical modeling and simulation are important research and monitoring tools used to understand biological communities and their relationships to the environment. Mathematical models are collections of variables, equations, and starting values that form a cohesive representation of a process or behavior. Because interactions among the members of biological communities and components of the abiotic environment are extremely complex, mathematical models are useful for understanding how ecosystems function and for making predictions about managing ecosystems.

There are numerous types of mathematical models used by ecologists and environmental scientists. Some models isolate the key factors that drive elements of a system. Other mathematical models are comprehensive simulations that include as many components and interactions as possible. Mathematical models also cover different spatial and temporal scales: from the smallest tidepool ecosystem to the entire planet; from a single day to millions of years.

for more information see for example

https://www.encyclopedia.com/environment/energy-government-and-defense-magazines/mathematical-modeling-and-simulation

I would say that it depends on how the output will be used. If the simulation is a part of - say - an optimization model then you need to make sure the simulation is a good enough representation of the real-world scenario - and to ensure that is not easy - such that if you are aiming to optimize some score, you will know roughly how reliable your simulation is for that specific purpose.

I would say that it depends on how the output will be used. If the simulation is a part of - say - an optimization model then you need to make sure the simulation is a good enough

@ Maged Gumaan Bin-Saad and @ Michael Patriksson Thanks for providing valuable comments. Can you specify how i should judge the mathematical model, if simulation results differ, Is it fair enough to be used in my scenario that i ahave mentioned initially in my question.

My simulation will not optimize model, rather it produces result for certain network algorithm in ns-3 simulator and i used mathematical model (equations) already derived for this algorithm by a researcher .

Математика это элемент культуры, мировоззрения и инструмент в повседневной жизни. Допустимая точность математических моделей определена автоматически областью и потребностями их применения.

Hi, please find attached a nice document that would help you figure out the relation between mathematical models and numerical simulations.

A mathematics model is a set of equations describing a phenomenon in nature and maybe even in the social sciences. It often and in fact usually involves a dynamical component (evolution over time). More often than not the models are approximations of the actual phenomenon. The approximations are made to make the model more mathematically tractable and any simulations more tractable. The approximations often involve choosing the important variables and disregarding others.

However, for example some of the predictions of the predator-prey equations (Lotka-Volterra equations) show that this approach is not always a reliable approach.

http://www.tiem.utk.edu/~gross/bioed/bealsmodules/predator-prey.html

As simple as the equations look and as pretty as the predictions - they don't always give the "truth." There are several reasons. Among them is nonlinear dynamical systems can have chaotic behavior and the behavior can bifurcated based on the parameters considered "not important."

https://www.hindawi.com/journals/ddns/2016/3565316/

In the early days of studying nonlinear dynamics it was hoped that structural stability was the norm. A system is structurally stable if the trajectories are unaffected by small perturbations (C1 perturbations to be exact) in the dynamics (the undying vector field defining the dynamical flow). It is in one and two dimensions. It was shown (Andronov and Pontryagin) that in dimensions of one the structurally stable vector fields form an open dense set in all vector fields. So one could approximate any dynamics by one that was structurally stable. Peixoton's theorem established this for two dimensional manifolds. Structural stability provides justification for applying the quantitative theory of dynamic systems to analyze concrete problems which was quite active to the late 1960's.

After the development of the Hartman-Grobman theorem Steve Smale attacked the problem of characterizing hyperbolic critical points under the assumption that structural stability was the norm in all dimensions. However, that was not to be as he found counter examples and they abound in dimension three. That means that in dimensions greater than 2 the structurally stable systems are not dense in the space of all vector fields defining a dynamical system (or in their equivalent interactive map on a manifold) and can not be used as models.

This shows up in the existence of chaotic systems, bifurcations form one state to a radically different state based on a parameter, etc. This also calls into question how well models based on only the "important" parameters are in predicting the behavior of actual systems. This is particularly true in cases were the model is developed only on a few "important" parameters.

There is a good non-mathematical discussion of this in "Infinite Powers" by Steven Strogatz where he discusses the history of how calculus and dynamical systems evolved to attack the problems of physics and the sciences. In particular there is a discussion on the dimensionality issues faced in the biological sciences where the number of variables needed to really understand the dynamics is quite high and all have to be considered in the dynamics. He discusses how the approach was used to develop the three drug approach to keeping the HIV virus in check.

https://www.amazon.com/Infinite-Powers-Calculus-Reveals-Universe/dp/1328879984/ref=sr_1_1?keywords=infinite+powers&qid=1582841996&s=books&sr=1-1

So the answer your original question - it all depends. If there is significant difference (and a factor of 100 seems significant unless there is a programming error in the simulation) then one has to wonder if the model itself is not itself unstable and prone to bifurcations and/or chaotic behavior. In any case it is questionable the model it is an accurate description of the underlying phenomenon is is trying to describe.

Thanks @Truman Prevatt for detailed answer,one of the points I understood is that if difference in model values are less than 100 percent then it's acceptable. What if I am having similar trend in the values, observable through plotting, and difference in values are greater than 100 percent.

Masood Abbasi,

Maybe I misunderstood your question. Normally the simulation is based on the model - namely it numerically integrates or solves the equations of the model. In this case the model should agree with the model. If this is the case then the fact that they disagree by such a large factor points to a programming error or stability issues within the model or stability issues with the numerical algorithms used to make the calculations.

If on the other hand if what you mean by simulation is a real world experiment where you count the collisions and then compare that to what the model predicts - then I would suspect your model might need some work. However, even in this case if you do not state the standard deviation of the experiment so it is difficult to determine how unlikely the model prediction is compared with the experiment (simulation) without a standard deviation.

If you could provide a little more detail and be a little more specific, I would be helpful.

Whichever the case below still applies as a general principle.

Every model is based on a set of mathematical equations. The parameters of this equations are estimates of the true parameters driving the system the model is designed to predict. Take the classical "three body problem" where the gradational focus of three bodies are used to estimate the positions of the three bodies over time. Say take the earth, moon and sun. Unlike the two body problem (two dimensions) the three body problem has no closed form solution. It is also not structurally stable and hence undergoes chaotic behavior. That translates to very small errors in the model parameters can lead to large errors in the behavior.

For example, when this is applied to earth satellites, the three bodies are the earth, the satellite and the moon. One can take the initial conditions of the satellite and then predict the position, velocity and acceleration of the satellite in the future. That is known as predicted ephemeris. However, the predicted ephemeris will age and over a short period will be insufficient for the use. The period is determined by the use of the satellite. The first attempt to address this issue was to recognize that relativistic effects require that the model employ both general and special relativity. Even at that the ephemeris gets stale and the error between predicted and actual becomes progressively larger. One reason is such effects of residual gradational effects of other planets and the sun are not accounted for and if they are accounted for the models become too unstable and chaotic in nature. The second thing is there is no accounting for such things as solar wind that will modulation the spacecraft's velocity hence change the dynamics.

The remedy is a feed back approach where tracking information is continually generated and fed back into the ephemeris generation model maintain an updated model that minimizes the location error of the predicted vs. observed. One of the classic examples of the extended Kalman filter for example is orbital determination of earth space satellites. That is the equations of motion are continually updated so that the predicted ephemeris is not stable.

https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=9&cad=rja&uact=8&ved=2ahUKEwikw_-C6P7nAhWOc98KHcweCMYQFjAIegQIChAB&url=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC6427518%2F&usg=AOvVaw1azMqEY3WQx1DlzhXMz6BX

For example the US NAVSTAR Global Positioning System ground segment consist of 16 tracking stations that report their ranging data in real time back to the master control station in Colorado, USA for real time ephemeris update. That is required to maintain the estimates of the satellite state vectors to meet the system accuracy requirements.

The key to any model is to insure that the mathematics is correct. Next is that the mathematics is a reasonable approximation of the actual phenomenon being modeled. Then the next step is to analyze the model to insure that it describes stable dynamics - for complex models it most likely won't if the dimension is large enough. Next is to determine that how the evolution of the model compares with the evolution of the actual system being modeled. If it is not within a reasonable error bound there are two possibilities. First there are important parameters that have been ignored in the model. Second there may be random disturbances in the systems - such as solar wind in orbital determination.

Refine the model with the new parameters. If in fact there is randomness, then some worth of probabilistic model will be required in which case the equations become stochastic and the solutions are probability distributions that can be tested.

it seems to me that you are at the step of evaluating if you model is sufficient to estimate the system you are trying to model. That is the beginning of the process. Better orbital determination models have evolved over time sine the first satellite that require fairly precise ephemeris generation was launched by the US Navy in 1960. The algorithms keep improving as we better understand all the factors going into the dynamics.