Can anyone help with the inverse problem and tuning parameters

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this picture summarises the model, the parameters and the data of the population of london from 1985 to 2015For my final year project i want to model the population of London using the Verhulst logistic model. However, to gain more marks i wish to use the inverse problem to tune the parameters to make the model match the real live values better. The problem is i have limited resources on the inverse problem and hope someone can point me in the right direction. Thankyou.

Also, if someone can recommend a better population model i would highly appreciate that aswell.