Directing us to move with stealth into a lot overgrown with poison oak vines and blackberry brambles, she pointed to a ring of eucalyptus trees. The morning had begun cold but the temperature had inched past 54 degrees, when monarchs begin to emerge from their slumber. The brown and green branches of one tree were stirring, as if a slight breeze was ruffling the dun-colored leaves. But then a distinctive orange color revealed itself. Butterflies peeled off from the branches, each one opening like a warm kiss before fluttering into the air.

Increasingly, people without formal backgrounds in science are collaborating with scientists to collect data on a scale that scientists alone would be unable to compile. The work of these people in recording the exact time, place and conditions of their butterfly observations is vital to monitoring the health of monarch populations. Tracking these butterflies is one of the longest-standing examples of this kind of teamwork.

Over the period of a year, monarchs produce four to five generations. The last and longest-lasting of them is born between August and October. Unlike their predecessors, which live as butterflies for a mere two to four weeks, these monarchs survive for six to eight months. After staying put over the winter in Mexico or California, they disperse in March or April, spreading far and wide in search of milkweed upon which to lay their eggs, which will morph into caterpillars that become the next generation of butterflies. The final generation in this yearlong cycle will return to the same California coast as their ancestors did. How these butterflies find their place of origin remains a mystery.

Last year’s count in Bolinas had been very low; still, the trees had been festooned with scores of butterfly clumps, in which hundreds of monarchs hung together for warmth and protection. This time, there was just one clump. Later we would learn that the total count of this site in Bolinas, which the previous year tallied 12,360 butterflies, plummeted this winter to just 1,256 monarchs. “This animal story that has been going on for centuries and perhaps thousands of years is disappearing and may be gone” soon, Ms. Monroe told us, her eyes tearing.

The total number of West Coast monarchs was estimated at approximately 4.5 million in the 1980s. In the latest count , that number fell to 28,429, dipping below the number scientists estimate is needed to keep the population going. This drastic decline indicates the migration is collapsing. The United States Fish and Wildlife Service is expected to announce in June whether its scientists think the monarch qualifies for protection under the Endangered Species Act.