This answer is based on the law of canada, which follows england-and-wales regarding the lack of a duty to investigate.
"It has been held many times that the police do not owe either a public law or a private law duty to any individual to investigate crime" (Holmes v. White, 2014 ONSC 5809, 2014 ONSC 5809, at para 16).
there is now a long list of decisions rejecting the proposition that the police owe victims of crime and their families a private law duty of care in relation to the investigation of alleged crimes [list of citations omitted]
(Wellington v. Ontario, 2011 ONCA 274, at para 20)
Nor is there any public aspect of the duty that courts can compel the police to exercise in a particular way.
Canadian law follows English law on this point:
The applicant relies heavily on Blackburn and cases which have followed it in England to try to argue that mandamus can lie to compel police to investigate particular crimes. Yet, Lord Edmund-Davies in Blackburn (at page 912, post F) and Lord Denning himself in R. v. Chief Constable of Devon and Cornwall, Ex p. Central Electricity Generating Board, [1982] 1 Q.B. 458 (CA) (at page 472) both said expressly the opposite. In both of those cases, the court required police to ignore a policy that purported to prevent them from even considering whether to investigate certain crimes regardless of how the police thought they should act. Once the policies, which the courts found to be unlawful, were swept aside, the courts then required the police to make the discretionary decision of whether or not to investigate the allegations.
The English cases concluded that while the court can require the police to exercise their discretion to decide whether to investigate, the court should not dictate to the police the outcome of that discretionary decision.
...
... [The applicant] submits that, taken together, these cases provide that every alleged victim of a crime has a right to have the court compel the police to either investigate his or her allegations or to require the police to establish in court that they have objectively and subjectively reasonable grounds to decline to investigate. Just stating the proposition is to reject it.
(Holmes v. White, 2014 ONSC 5809, at paras 17-20)
However, investigating officers do owe a duty of care to a suspect in the course of an investigation. The standard requires police officers to exercise their discretion within the bounds of reasonableness owed to the suspect. This is an outer bound on the aggressiveness or manner or choice of tactics used in investigating a suspect. During an investigation, the standard of care requires police to act within the bounds of reasonableness, defined as the conduct of a reasonable police officer in similar circumstances.
At the outset of an investigation, the police may have little more than hearsay, suspicion and a hunch. What is required is that they act as a reasonable investigating officer would in those circumstances.
However, I emphasize it is a duty owed to the suspect, not the complainant, and it does not require any investigation at all. See generally Hill v. Hamilton‑Wentworth Regional Police Services Board, [2007] 3 S.C.R. 129, 2007 SCC 41.