This paper deals with an optimal distribution of 
search effort for a target moving on two-dimensional 
search space, say a geographical space with time flow. 
In most researches published so far on this subject, 
the amount of available search effort has an upper 
limit only at each time so that the algorithm of 
repeatedly solving a kind of resource allocation problem 
at each time worked well to obtain optimal solutions. 
However, on the two-dimensional search space, we have to 
consider the nested constraints of search effort, which 
mean constraints on the total amount of available effort 
on the whole space as well as at each time and make the 
problem difficult to be solved. In this paper, we derive 
necessary and sufficient conditions for optimality and 
propose two algorithms for an optimal solution, which 
perform better than some well-known nonlinear programming 
methods in terms of computational time.